Index

general.doc / A quick guide to Emacs

Tutorial-style introduction to Emacs for Singular users, focusing on basic concepts and common commands. Includes key abbreviations and essential window, buffer, help, and exit operations.

classify.lib

Return the Arnold normal form for a polynomial or for a named singularity from the database.

general.lib

Return a comma-separated list of n alphabet letters starting at s, in alphabetical (or reverse) order.

general.lib

Build a list of consecutive letters starting from a given letter, or indexed names for longer sequences.

KVequiv.lib

Returns the absolute value of an integer.

2 commands

A library for computing the absolute factorization of multivariate polynomials f with coefficients in a field K of characteristic zero.

absfact.lib

Compute the absolute factorization of p and return a new ring containing a list absolute_factors.

absfact.lib

Absolute factorization of a polynomial by adjoining one new parameter; returns a new ring with list absolute_factors.

primdec.lib

Compute absolute prime components of an ideal (via GTZ + absFactorize) and return a ring containing lists absolute_primes and primary_decomp.

primdec.lib

Compute absolute prime components of an ideal (char 0) and return a ring with lists absolute_primes and primary_decomp (primdecGTZ output).

reszeta.lib

Convert an embedded resolution (resolve output) to data describing the corresponding non-embedded resolution for isolated surface singularities.

general.lib

Returns the absolute value of c (c if c>=0, otherwise -c) for types supporting comparisons.

types.doc / Access to elements of a user defined type

Explains how to access elements of a user-defined type using . and describes ring-dependent element names. Includes a short Singular example.

ainvar.lib

Test whether the group action defined as exp of a vector field given by an n×1 matrix is proper.

redcgs.lib

Builds term and coefficient ideals from m, taking coefficients from m1 when row-1 entries match, else 0; returns [T,C].

redcgs.lib

Recursively adds/refines cases in the global tree @L for node labu using specification data rs; returns 1/0 status.

matrix.lib

Return A after adding p times column c1 to column c2.

resources.lib

Adjust the number of available processor cores by adding an integer offset and return the new core count.

sets.lib

Return the set M with element a added.

methods.lib

Join two hash tables by concatenating keys and values (assumes disjoint key sets; not checked).

olga.lib

Add two left fractions in a specified localization (given by locType and locData) and return the sum as a fraction vector.

methods.lib

Combine two Method objects by merging their functionClosures into a new Method.

modules.lib

Return the sum of two modules (submodules of the same ambient module).

normaliz.lib

Add a custom Normaliz option mapping (shortcut to option); the option is registered but not activated.

realclassify.lib

Adds signed squares of non-degenerate variables to the normal form equation stored in a NormalFormEquation.

ring.lib

Add N variables to a ring (extending it with newly named variables).

polyclass.lib

Add two polynomials of type Poly.

stanleyreisner.lib

Return the sum of two QPoly elements in their quotient ring (computed as the normal form of f+g).

dmodloc.lib

Add two rational functions given as 2-component vectors [num,den] and return the resulting [num,den] vector.

matrix.lib

Return matrix A with row r2 replaced by row r2 + prow r1.

schubert.lib

Return the direct sum of two sheaves.

ring.lib

Add variables to a ring.

fpalgebras.lib

Create a ring in characteristic 0 and export the ideal generator AdemRel for the Adem relation with 0<i<2j.

grobcov.lib

Computes supplementary parameter conditions so (H and not H1) => (T and not T1) becomes a theorem in Q[a][x].

brnoeth.lib

Compute fundamental data for a plane curve (incl. degree, genus, places) and its conductor/adjunction divisor.

linalg.lib

Compute the adjoint (adjugate) of a square matrix A such that AdjA = det(A)E.

paraplanecurves.lib

Compute the adjoint ideal of a plane curve defined by a homogeneous absolutely irreducible polynomial in three variables.

interval.lib

Compute the adjunct (adjugate) of an interval matrix as the transpose of its cofactor matrix.

ncpreim.lib

Check whether variables indexed by an intvec form an admissible subalgebra (returns 1 if yes, else 0).

gitfan.lib

Compute all a-faces as 0/1 intvecs; optionally filter by a given list of candidate faces or by a minimum face size.

examples.doc / AG codes

Describes the brnoeth.lib library for solving the Riemann–Roch problem via the Brill–Noether algorithm and its application to algebraic geometry codes, with a worked example in Singular.

tutor.tex / AG codes

This section introduces the use of brnoeth.lib for solving the Riemann–Roch problem and computing and decoding Algebraic Geometry codes on plane curves over prime fields.

brnoeth.lib

Return a generator matrix of the evaluation AG code defined by divisors G and D on the extended curve data EC.

brnoeth.lib

Return a generator matrix of the residual AG code defined by divisors G and D from extcurve data EC.

9 commands

## Invariant Rings of the Additive Group

8 commands

Algorithms for primality testing in polynomial time, based on the ideas of Agrawal, Saxena, and Kayal.

alexpoly.lib

Compute Alexander polynomial (and monodromy zeta) of a plane curve singularity; returns a list whose first entry is a ring containing alexpoly, zeta_monodromy, alexfactors.

14 commands

A library for computing invariants of a plane curve singularity f: the resolution graph of a minimal good resolution, total (and strict) multiplicities of the total transforms of the branches along the exceptional divisors, the Alexander polynomial, and the

algebra.lib

Compute the kernel of a map as a string; optionally create/export the kernel ideal in the preimage ring using a chosen algorithm.

algebra.lib

Test algebraic dependence of generators f1,...,fm and return a flag plus a ring containing the relation ideal ker in y(1..m).

algebra.lib

Test whether a polynomial lies in the subalgebra generated by an ideal; optionally return a witness ring and polynomial.

12 commands

This library provides procedures to compute with subalgebras and with algebra (ring) maps, including containment queries, algebraic dependencies, kernels, basic map properties (injective/surjective/bijective), Noether normalization, and finiteness tests.

examples.doc / Algebraic dependence

Explains how to test algebraic dependence of polynomials and subring membership using elimination and normal forms in an extended polynomial ring.

tutor.tex / Algebraic dependence

Explains how to test algebraic dependence of polynomials and subring membership using elimination and normal forms in an auxiliary ring.

sagbi.lib

Return a ring containing ideal algDep with (possible) algebraic dependencies found from I after given SAGBI iterations.

math.doc / Algorithms

Lists the algorithms implemented in the Singular library toric_lib, with links to the corresponding documentation nodes.

kernel

kernel

reference.doc / align

Describes the Singular command align, which shifts module generators by a given integer. Includes syntax, type, purpose, and examples.

purityfiltration.lib

Compute all double Ext-modules Ext^i(Ext^j(M,D),D) for the left module M=coker(R) over the current ring D.

purityfiltration.lib

Compute a list of matrices presenting the right Ext^i_D(M,D) modules for a left module M=coker(Ps).

purityfiltration.lib

Compute a list of matrices representing left Ext-modules Ext^i(M,D) for a right module M=coker(R).

nchomolog.lib

Compute and print Ext^i modules of N for i=1..nvars(basering); optionally use an involution defined by opts[1].

normaliz.lib

Print a list of available Normaliz options in this library (name, default value, and option passed to Normaliz).

bfun.lib

Return 1 if all components of an intvec are positive, otherwise 0.

inout.lib

Prints list elements if global int ALLprint is defined; pauses when ALLprint>0.

rootsur.lib

Test whether all roots of a univariate polynomial are real (returns 1 iff all real, else 0).

rootsur.lib

Test via Sturm's theorem whether all roots of a univariate rational-coefficient polynomial are real.

surfex.lib

Return a list of strings with all real roots of the active ring's minimal polynomial.

hnoether.lib

Compute the greatest squarefree divisor of f, completing missing factors not present in g=squarefree(f) via factorization.

nchomolog.lib

Computes an involution-based Ext module (or formal adjoint for i=0) from a matrix presentation using a resolution.

nctools.lib

Returns the index (number) of the last alternating variable in the current basering (a super-commutative algebra).

nctools.lib

Returns the index (number) of the first alternating variable of the current basering (must be a super-commutative algebra).

gfan.lib

Return the dimension of the ambient space in which a cone, fan, or polytope lives.

memory/OMALLOC.texi

Short overview of omalloc, Singular’s internal memory manager, including design principles, requirements, and performance/status notes.

primdec.lib

Compute the annihilator ideal of the cokernel R^m/M for a module M over the current basering.

dmodideal.lib

Compute the annihilator of f^alpha in D (OT99 method) for commutative char 0 basering; result ring contains annfalpha.

dmod.lib

Compute the D-module structure of basering[1/f]f^s and Bernstein-Sato data (LD, BS) in a returned output ring.

dmod.lib

Compute Ann(f^s0) in the Weyl algebra and Bernstein polynomial root data from a known Ann(f^s) ideal in D[s].

dmod.lib

Compute Ann(f^s) in the Weyl algebra and Bernstein data from a known Ann(f^s) in D[s], returning a ring containing LD and BS.

dmod.lib

Constructs a ring encoding the D-module structure of basering[1/f]f^s via the Briancon–Maisonobe algorithm.

dmod.lib

Compute annihilator of F[1]^s1...F[P]^sP and a Bernstein-Sato ideal for f=F[1]...F[P] (Briancon-Maisonobe).

dmodideal.lib

Construct an output ring containing annfsLog, the (non-GB) logarithmic annihilator ideal of a multi-factor F^s in D.

dmod.lib

Compute (and return as a ring) the D-module structure of basering[1/f]f^s using the Levandovskyy-modified Oaku–Takayama algorithm.

dmod.lib

Compute (Oaku–Takayama) the D-module structure of basering[1/f]f^s; returns an output ring containing LD and BS.

dmod.lib

Compute the generic annihilator Ann(F^s) for a parametric polynomial (Briancon–Maisonobe) and the exceptional parameter conditions.

dmod.lib

Compute the annihilator ideal of F^n in the Weyl Algebra for a given polynomial F and rational number n, returning an output ring.

dmod.lib

Compute the annihilator ideal of F^n in the Weyl algebra from Ann(F^s) in D[s], for a rational exponent n.

dmod.lib

Compute Ann(f^s0) in the Weyl algebra and Bernstein polynomial roots from a known Ann(f^s) in D[s], using the Jacobian ideal.

dmodideal.lib

Compute the annihilator ideal Ann(F[1]^s1...F[P]^sP) in a returned D-module ring (Briancon–Maisonobe algorithm).

mprimdec.lib

Compute a standard basis ideal for the annihilator of the quotient module M/N in the basering.

dmodapp.lib

Compute the annihilator ideal of a polynomial f in the corresponding Weyl algebra, returning that Weyl ring with ideal LD exported.

dmodapp.lib

Compute the annihilator ideal (LD) of the rational function g/f in the corresponding Weyl algebra.

dmodloc.lib

Constructs a Weyl-algebra ring containing an ideal LD giving (part of) the annihilator of the rational function g/f.

dmodapp.lib

Create a parametric Weyl algebra ring containing the ideal IAppel1 annihilating the Appel F1 hypergeometric function.

dmodapp.lib

Create a parametric Weyl algebra ring containing the ideal IAppel2 that annihilates the Appel F2 hypergeometric function.

dmodapp.lib

Create a parametric Weyl algebra ring containing the ideal IAppel4 that annihilates the Appel F4 hypergeometric function.

ssi.lib

Append a list of expressions/objects to an SSI file at the given path.

ncpreim.lib

Return the basering with an appended weight ordering component (a(w)) in front of its existing ordering.

examples.doc / Applications

Links to example applications, including solving polynomial systems and AG codes.

tutor.tex / Applications

Entry points to application-oriented topics in the tutorial, including solving polynomial systems and AG codes.

reference.doc / apply

Applies a function to all elements of a vector, matrix, or list. Returns an expression list whose type and format are determined by the subsequent assignment.

central.lib

Apply the adjoint map Ad_f(v)=fv-vf to each generator of an ideal and return the generated ideal.

interval.lib

Apply an interval matrix to a box, returning the product Ab.

gitfan.lib

Applies a permutation to the entries of an intvec and returns the resulting intvec sorted.

VecField.lib

Apply a VecField V as a differential operator to a polynomial or to each generator of an ideal, optionally truncating V to degree n.

qhmoduli.lib

Substitute an ideal into a polynomial, drop terms above a weighted-degree bound, then reduce modulo a given ideal.

arc.lib

Create and return a ring containing the ideal arcideal for the arc space of (I at fat point J), optionally skipping coefficient indices.

arc.lib

Internal arc.lib routine operating on ideals I and J (nonzero) with optional settings via optList/useOptList.

arc.lib

Construct an arc ring for scheme I at fat point J, optionally skipping selected coefficient indices via optList.

3 commands

Computes the equations of a generalized arc space of a scheme defined by the ideal I, relative to a fat point defined by the ideal J. Optionally, a skip list of coefficients can be provided.

5 commands

An arc is given by a power series in one variable, say \(t\). Truncating it at a positive integer \(i\) means deleting all terms with \(t\)-powers \(> i\). The set of arcs truncated at order \(<\) bound is denoted \(\mathrm{Tr}(i)\).

monomialideal.lib

Compares two lists of monomial ideals by multiplying each ideal's generators and testing equality of the resulting ideals.

olga.lib

Check whether two given (left) fractions are equal in the localization specified by locType and locData.

multigrading.lib

Check whether all column vectors of an integer matrix are zero elements in the (multi)grading group.

36 commands

This library classifies isolated hypersurface singularities at the origin which are (right- and stably) equivalent to a germ with a nondegenerate Newton boundary of corank \(\le 2\) (in the sense of Kouchnirenko).

qhmoduli.lib

Compute the induced action of the stabilizer G_f of a quasihomogeneous f on T_ (upper monomials), returning a ring with actionid and stabid.

arnoldclassify.lib

Classify an isolated hypersurface singularity (Arnol'd list, corank ≤ 2) up to right equivalence and return invariants/normal form.

arnoldclassify.lib

Classify an Arnol'd corank≤2 isolated singularity and return invariants and normal form as a '|' separated string.

9 commands

A library for classifying isolated hypersurface singularities from the list of V. I. Arnol'd with respect to right equivalence up to corank 2. The method relies on Baciu's list of Milnor codes and Newton's rotating ruler method to distinguish the Y- and Z-sing

arnoldclassify.lib

Compute the corank of the Hessian matrix of a polynomial f (at the origin) in a local basering.

arnoldclassify.lib

Computes an upper bound for the right determinacy of a singularity, returning -1 if infinite.

qhmoduli.lib

Compute the induced action of G_f on T_ from a quasihomogeneous f and its semiuniversal unfolding Fs.

arnoldclassify.lib

Return a list of names of Arnol'd singularity series up to corank 2.

arnoldclassify.lib

Compute the Milnor code (as an intvec) of a singularity f from the 2nd Hilbert series of basering/(jacob(f)^e).

arnoldclassify.lib

Apply the Splitting Lemma to f, returning a polynomial g in maxideal(3) right equivalent to f.

arnoldclassify.lib

Compute the Arnol'd normal form for a singularity series or class, optionally with integer and polynomial parameters.

arnoldclassify.lib

Return database data for a named Arnold singularity series as a singseries object.

arnold.lib

Return the Arnold classification string stored in N.ArnoldType for a NormalForm (computes it via determineArnoldType if needed).

arr.lib

An arrangement of hyperplanes.

44 commands

## arr.lib

arr.lib

Convert an arrangement to a multiarrangement by assigning multiplicities from an intvec.

dmod.lib

Construct the product polynomial defining a full hyperplane arrangement in the current commutative basering.

JMSConst.lib

Group a degree-sorted monomial ideal into sublists by degree (for J-marked scheme computations).

JMSConst.lib

Constructs the final list of J-marked polynomials by inserting empty degree blocks for missing intermediate degrees.

arr.lib

Construct the boolean arrangement arr using the first v variables of the current ring.

arr.lib

Compute the number of bounded chambers of an arrangement (Poincaré polynomial evaluated at -1).

arr.lib

Construct the Type A (braid) hyperplane arrangement of dimension v (hyperplanes x(i)=x(j)).

arr.lib

Computes the center (common intersection) of an arrangement by solving the associated linear system.

arr.lib

Check whether an arrangement is centered (i.e. the intersection of all hyperplanes is nonempty).

arr.lib

Check whether an arrangement is central (all planes intersect in 0).

arr.lib

Make a centered arrangement central by applying a translational coordinate change.

arr.lib

Return the number of chambers of an arrangement (Poincare polynomial evaluated at 1).

arr.lib

Return coefficients of the characteristic polynomial of an arrangement (in increasing order).

arr.lib

Return the coned arrangement by homogenizing with respect to a given (or last) ring variable and adding that variable as a new hyperplane.

arr.lib

Performs an invertible coordinate change x↦Tx+c on an arrangement, updating [A|b] to [AT^-1|b+AT^-1c].

arr.lib

Normalize an arrangement by a coordinate/affine change so selected rows become identity and selected constants become zero.

arr.lib

Decone a non-empty central hyperplane arrangement with respect to the k-th hyperplane.

arr.lib

Compute the module Der(A) of derivations tangent to the (multi-)arrangement A (central only).

arr.lib

Construct the Edelman–Reiner hyperplane arrangement (requires the active ring to have at least 5 variables).

arr.lib

Return an essential arrangement obtained from a non-empty arrangement by variable transformations.

arr.lib

Return the exponents (degrees of a basis of the derivation module D(A)) of a free central (multi-)arrangement.

arr.lib

Compute the intersection lattice (as an arrposet) of an arrangement.

arr.lib

Check whether an arrangement is essential (rank of maximal element equals the dimension).

arr.lib

Test whether a (multi-)arrangement is free (Der(A) is a free module); returns 1 if free, else 0.

arr.lib

Return the index (as int) of the last/largest ring variable used in an arrangement.

arr.lib

Computes the intersection lattice/poset of a hyperplane arrangement.

arr.lib

Localize an arrangement onto a flat defined by A[v], keeping only hyperplanes containing that flat.

arr.lib

Construct the exterior algebra for an arrangement and attach the Orlik–Solomon ideal I as a ring attribute.

arr.lib

Compute the Poincaré polynomial of a hyperplane arrangement as an integer vector.

arr.lib

A poset representing the intersection lattice of an arrangement.

arr.lib

Print an arr in readable matrix form, multiplying coefficient columns by the corresponding variables.

arr.lib

Create a random (affine) hyperplane arrangement with bounded integer coefficients.

arr.lib

Create a random central hyperplane arrangement with bounded integer coefficients.

arr.lib

Restricted arrangement onto a flat.

arr.lib

Replace the k-th hyperplane in an arrangement by a given linear polynomial.

arr.lib

Swap two variables in an arrangement A (by ring variables or by integer indices into variables(A)).

arr.lib

Construct the type B reflection arrangement using the first v variables of the current ring.

arr.lib

Construct the type D reflection arrangement using the first v variables of the current ring.

general.lib

Return printable ASCII characters: full table (no args), a single character by code, or a range by codes.

types.doc / ASCII links

Describes how ASCII links write data as plain ASCII for storage or communication, and how reading returns strings. Explains link string formats, default behaviors, and a usage example.

aksaka.lib

Deterministic primality test (ASK algorithm): returns 1 if n is prime, 0 if composite.

ehv.lib

Compute the associated prime ideals of an ideal using the Eisenbud/Huneke/Vasconcelos algorithm.

types.doc / Assignments for user defined types

Explains how assignments work for user-defined types in Singular and how to override = to implement custom constructors and generic type conversions.

grobcov.lib

Given a family F with constraints C and an envelope component E, compute the associated tangent hypersurface parameters at a point of E.

ehv.lib

Return an ideal whose associated primes are exactly the associated primes of I having dimension e.

assprimeszerodim.lib

Computes the associated primes of an ideal I.

2 commands

A library for computing the radical and the associated primes of a zero-dimensional ideal in the polynomial ring over the rational numbers, \(\mathbb{Q}[x_1,\ldots,x_n]\), using modular computations.

general.doc / ASSUME

Describes the ASSUME command, which conditionally tests an expression based on an assumeLevel and raises an error if the test fails.

kernel

attribute object

reference.doc / attrib

Documents the attrib command for displaying, retrieving, and setting attributes on Singular objects. Includes notes on reserved attributes used by the kernel and libraries.

autgradalg.lib

Compute generators of the subgroup of GL(k,ZZ) that permutes/fixes the set of generator degrees given by the columns of Q.

autgradalg.lib

Compute the automorphism group of a multigraded algebra R/I as an algebraic subgroup, returning a ring and exporting Jexported and stabExported.

6 commands

This library provides a framework for computing automorphisms of integral, finitely generated algebras that are graded by a finitely generated abelian group. It also contains procedures to compute automorphism groups of Mori dream spaces.

autgradalg.lib

Compute Aut_K(S) of a multigraded polynomial basering as an algebraic subgroup V(Iexported) of GL(n), returning the coordinate ring.

control.lib

Investigate autonomy-related properties of a system (behavior) represented by the module/matrix R.

control.lib

Compute autonomy-related properties (e.g., autonomy/strong autonomy, dimension, degree, kernel representation, determinacy) for a system matrix R.

autgradalg.lib

Compute generators for the Hopf algebra O(Aut(X)) of X(R,w); returns a ring and exports the ideal Iexported.

autgradalg.lib

Compute a ring/ideal description of the H-equivariant automorphism group of widehat X as an algebraic subgroup of some GL(n).

COPYING.texi

License and availability information for Singular, including bundled modules, optional external programs, and acknowledgements of developers and supporting institutions.

grwalk.lib

Compute a standard basis via the first alternative Groebner walk algorithm between specified weight orderings (with optional perturbation degree).

grwalk.lib

Compute a transformed Groebner basis via MAltwalk1 using an algebraic-lp style ordering derived from args.

grwalk.lib

Compute a standard basis of an ideal using alternative Groebner walk algorithm 2 between weighted (a(),lp) orderings.

crypto.lib

Compute the discrete logarithm x such that b^x ≡ y (mod p) using Shanks' baby-step giant-step method.

start.doc / Background

Overview of Singular, its core computational objects and algorithms, and a brief history of its development and releases.

tutor.tex / Background

Overview of Singular and its main computational focus, features, and development history. Includes a pointer to version-specific highlights.

finitediff.lib

Return a predefined backward finite-difference approximation for two derivative names with respect to a basering variable.

reference.doc / bareiss

Applies the sparse Gauss–Bareiss algorithm with an optimal pivot strategy to a module (or a matrix). Returns a list containing a reduced module and an intvec describing the row permutation.

compregb.lib

Pretty-print the result object G as a string representation.

compregb.lib

Convert a list of bases to a single string by applying basis2str to each element and joining with newlines.

surfex.lib

Return the RGB triple (as a list of three integers) for a base surface color selected by index modulo numBaseColors().

examples.doc / Basic programming

An example-based introduction to basic programming in Singular. It demonstrates ring setup, ideals, Gröbner bases, dimensions, primary decomposition, localization, and coordinate transformations.

tutor.tex / Basic programming

Example-driven introduction to basic programming tasks in Singular, including rings, ideals, Gröbner bases, dimensions, strings, libraries, primary decomposition, and local computations.

classify.lib

Compute an upper determinacy bound, Milnor number and corank of a polynomial.

compregb.lib

Convert a 2-part basis description list into a formatted string with factor and polynomial conditions plus a variable list.

fpalgebras.lib

Constructs a Letterplace/free algebra ring for the Baumslag group and exports the relation ideal I.

fpalgebras.lib

Constructs a free algebra ring (monoid/group variant) for Baumslag–Solitar relations and exports the defining ideal I.

tricks.doc / Behavior of continue

Describes where Singular’s continue is valid and how it behaves inside loops, including a key difference from C. Includes a short example demonstrating the behavior.

tateProdCplxNegGrad.lib

Compute the Beilinson bundle U^a (pullback from each projective-space factor) over the current basering.

tateProdCplxNegGrad.lib

Compute the Beilinson window subquotient complex of a multigraded complex, keeping summands generated in degrees 0<=a<=n.

cimonom.lib

Test whether n lies in the semigroup generated by v; optionally return a monomial witness using variables indexed by sup.

cisimplicial.lib

Find nonnegative integer coefficients c with Ac = v (optionally restricted to the first k columns), else return 0.

ffmodstd.lib

Compute (and optionally update) the minimal polynomial for a sequence via the Berlekamp–Massey algorithm.

chern.lib

Compute the list of (second) Bernoulli numbers B(0) through B(n) using the Akiyama–Tanigawa algorithm.

gmssing.lib

Compute the Bernstein–Sato polynomial b(s) of a polynomial t, returning its roots and multiplicities.

dmod.lib

Compute the global Bernstein–Sato polynomial of a hypersurface (Briançon–Maisonobe algorithm) and return its roots with multiplicities.

dmod.lib

Compute a candidate (multiple of) Bernstein–Sato polynomial of f via a lift-like method, returning roots with multiplicities.

dmodideal.lib

Compute Bernstein–Sato ideals (and annihilator data) for f=product(F[i]) via the Briancon–Maisonobe multivariate algorithm.

numerDecom.lib

Read a Bertini solutions file and return the computed homotopy solutions as a list of complex coordinate lists.

modules.lib

Return the Betti matrix of a given resolution.

reference.doc / betti

Describes the Singular command betti for computing graded Betti numbers from a (minimal) resolution of R^n/M. Includes syntax variants, notes on minimization, and an example.

plural.doc / betti BR_PLURAL_BR

Describes betti, which computes graded Betti numbers from a (minimal) resolution, and explains output format and options, including notes for the non-commutative case.

goettsche.lib

Compute Betti numbers of the Hilbert scheme of n points on a surface with given Betti numbers b.

goettsche.lib

Compute the Betti numbers of the moduli space of Kronecker modules N(q; m, n).

goettsche.lib

Compute the Betti numbers of the punctual Quot-scheme of rank r on n points on a plane.

goettsche.lib

Compute the Betti numbers of the n-th symmetric power of a variety from its Betti numbers list.

dmodapp.lib

Tries to compute roots of a univariate polynomial, returning roots as an ideal with multiplicities and optionally a leftover irreducible part.

dmodideal.lib

Compute upper/lower bounds for the Bernstein–Sato ideal for m via Budur (Bud12), returning a ring with Bj, shiftedIdeals, upperBound, lowerBound.

bfun.lib

Compute roots (and multiplicities) of the Bernstein–Sato polynomial b(s) for a hypersurface f using linear reductions.

bfun.lib

Compute roots (with multiplicities) of the Bernstein–Sato polynomial b(s) of a hypersurface f via Ann(f^s) (+ f/partials).

dmodloc.lib

Compute roots and multiplicities of a multiple of the b-function for f^su at a generic root of f.

bfun.lib

Compute the rational roots (and multiplicities) of the global b-function of an ideal w.r.t. the weight (-w,w).

bfun.lib

Compute roots (with multiplicities) of the Bernstein–Sato polynomial b(s) for f using a single Groebner basis computation.

bfun.lib

Compute roots of the Bernstein–Sato polynomial b(s) for a hypersurface f using a syzygy-based linear solver.

dmodvar.lib

Compute roots of the Bernstein–Sato polynomial b(s) of V(F) (via Sannfs), returning roots (ideal) and multiplicities (intvec).

dmodvar.lib

Compute rational roots of the Bernstein–Sato polynomial b(s) of V(I) and their multiplicities using an initial ideal approach.

plulibs.doc / bfun_lib

Given a polynomial ring \(R = K[x_1,\ldots,x_n]\) and a polynomial \(F \in R\), one is interested in the global b-function (also known as the Bernstein–Sato polynomial) \(b(s) \in K[s]\). It is defined as the non-zero monic polynomial of minimal degree satisfy

13 commands

Given a polynomial ring \(R = K[x_1,\ldots,x_n]\) and a polynomial \(F \in R\), one is interested in the global b-function (also known as the Bernstein–Sato polynomial) \(b(s) \in K[s]\). It is defined as the non-zero monic polynomial of minimal degree satisfy

References extracted from the documentation.

polybori.lib

PolyBoRi ideal type corresponding to Singular's @code{ideal}.

polybori.lib

Return the union of two bideal generator lists as a sorted, duplicate-free bideal.

polybori.lib

Convert a PolyBoRi boolean ideal (bideal) to a Singular ideal.

polybori.lib

Return the Python-object (pylist) representation stored in a Boolean ideal.

math.doc / Bigatti and La Scala and Robbiano

Describes the algorithm of Bigatti, La Scala and Robbiano and its relation to the algorithms of Pottier and of Hosten and Sturmfels.

kernel

big integer

types.doc / bigint

The type bigint represents arbitrarily large integers. Values can only be constructed from other types such as int and number.

types.doc / bigint declarations

Declares a long integer variable of type bigint, including syntax, default value, and an example.

types.doc / bigint expressions

Defines what constitutes a bigint expression in Singular and provides an example illustrating type casting to bigint. Includes references to related types and conversions.

types.doc / bigint operations

Lists the supported arithmetic and comparison operations for the bigint type, with a short example.

types.doc / bigint related functions

Lists functions related to the bigint type and provides cross-references to their documentation nodes.

kernel

Big integer matrices are matrices with big integer entries. No basering definition is required to use bigint matrices, for they do not belong to a ring. Bigintmat entries can have any size because of the use of bigint.

types.doc / bigintmat

Big integer matrices are matrices whose entries are big integers. This section introduces their properties and links to declarations, expressions, type casts, and operations.

types.doc / bigintmat declarations

Declares variables of type bigintmat, including sized matrices and initialization from bigintmat expressions or lists of (big) integers.

types.doc / bigintmat expressions

Describes what constitutes a bigintmat expression in Singular and provides a short example. Includes related references to bigintmat and type conversion/casting.

types.doc / bigintmat operations

Overview of available operations on bigintmat, including arithmetic, comparison, and element access, with a short example.

types.doc / bigintmat type cast

Describes the bigintmat type cast, including syntax, purpose, and an example. Also links to related documentation nodes.

gitfan.lib

Return positions of 1-bits in the r-bit binary representation of a bigint n.

kernel

Big integer vectors are vectors with big integer entries. No basering definition is required to use bigint vectors, for they do not belong to a ring. Bigintvec entries can have any size because of the use of bigint.

types.doc / bigintvec

Big integer vectors are vectors whose entries are arbitrary-size integers. They do not require a basering definition because they do not belong to a ring.

types.doc / bigintvec declarations

Describes the syntax for declaring bigintvec variables in Singular, including defaults and a short example.

types.doc / bigintvec expressions

Defines what constitutes a bigintvec expression in Singular and provides a short example.

types.doc / bigintvec operations

Operations and comparisons supported for the bigintvec type, with a short usage example.

types.doc / bigintvec type cast

Casts an expression to type bigintvec. The created bigintvec has the same size as the input expression.

letterplace.doc / Bimodules and syzygies and lifts

Introduces free bimodules over a free algebra, defines syzygy bimodules (bisyzygies), and describes lift(ing) as encoding generator expressions via matrices over the enveloping algebra.

plulibs.doc / bimodules_lib

The main purpose of this library is handling bimodules. This can help, for example, to determine weak normal forms of representation matrices and total divisors in non-commutative, non-simple G-algebras.

12 commands

The main purpose of this library is handling bimodules. This can help, for example, to determine weak normal forms of representation matrices and total divisors in non-commutative, non-simple G-algebras.

crypto.lib

Return the given binary-encoded list incremented by 1.

gitfan.lib

Convert a sparse binary index list (positions of 1s) to a bigint.

systhreads.lib

Bind a typed shared object with the given name as an atomic_list.

systhreads.lib

Bind a typed shared object named name as an atomic_table.

systhreads.lib

Bind a typed shared object named name as a channel.

systhreads.lib

Bind a typed shared object named name as a "region".

systhreads.lib

Bind a typed shared object named name as a "syncvar".

systhreads.lib

Bind a shared object by name only if its type matches the required type; otherwise signals an error.

general.lib

Compute the binomial coefficient (n choose k) for integer n,k.

normaliz.lib

Convert a binomial ideal to an intmat whose rows are exponent-difference vectors (length nvars(basering)) for each non-zero generator.

resbinomial.lib

Compute an E-resolution of singularities of a binomial ideal, returning the resolution data as a list of affine charts.

bimodules.lib

Compute a two-sided Groebner basis of a (two-sided) ideal or module using the enveloping algebra of the basering.

bimodules.lib

Compute bistd directly for an ideal (using std in the enveloping ring and mapping back).

bimodules.lib

Compute enveloping algebra ring for M and export L=[Coeff,Bisyz] (coefficient/lift matrix and bisyzygies for the two-sided Groebner basis).

bimodules.lib

Applies bitrinity to an ideal; constructs envelope/elimination data and exports coefficient and bisyzygy matrices.

curvepar.lib

Compute the blow-up of the curve V(I) at the origin, returning a list of affine charts with maps and transformed ideals.

resolve.lib

Compute the blowup of the ambient space V(W) in a smooth center V(C) and the strict transform of V(J), returning affine charts with data.

scheme.lib

Compute the blow-up of a scheme X along a subscheme C, returning (blow-up scheme, blow-up map, exceptional divisor).

elim.lib

Construct a presentation ring for the projective blowup in center C and compute exceptional divisor, total/strict transform of J, and blowup map.

resolve.lib

Compute the blow-up of V(J) along center C, returning chart rings with Jnew, exceptional divisor eD, and blowup map ideal bM.

resolve.lib

Compute the blow-up of a basic object/presentation along center ideal C and return the chart rings with transformed data.

resbinomial.lib

Performs the blow-up in a specified chart along a given center, returning the new affine charts and related data.

scheme.lib

A blow-up morphism together with associated data (base, blow-up, center, exceptional divisor) and the defining chartmaps.

scheme.lib

Convert a blowUpMap into the corresponding morphism.

kernel

kernel

types.doc / boolean expressions

Defines boolean expressions in Singular and lists available comparisons for different object types. Notes evaluation order for logical operators and differences from C-style short-circuiting.

types.doc / boolean operations

Lists the boolean operators and their alternative notations, and gives their precedence and examples.

polybori.lib

Convert an integer constant to a Boolean constant representation.

polybori.lib

Convert an ideal to the representation of a Boolean ring (optionally using a specified Boolean ring).

polybori.lib

Convert a polynomial to a Boolean polynomial representation.

polybori.lib

Create (and cache) a PolyBoRi boolean polynomial ring corresponding to the given Singular ring.

polybori.lib

Convert a ZDD into a PolyBoRi boolean set, optionally specifying the boolean ring.

polybori.lib

Compute the boolean Groebner basis of an ideal (adding x(i)^2-x(i) implicitly for Singular ideals).

rstandard.lib

Computes a border basis of the ideal I.

JMSConst.lib

Check whether a monomial ideal is Borel by testing all Borel moves reduce to zero modulo the ideal.

rootsur.lib

Returns a Budan–Fourier upper bound (correct parity) for the number of real roots of p in (a,b].

rootsur.lib

Compute an upper bound (with correct parity) for the number of real roots of a univariate rational polynomial.

rootsur.lib

Return an upper bound (with correct parity) for the number of positive real roots of a univariate rational polynomial.

rootisolation.lib

Create an interval with given bounds: [a,a] or [a,b].

interval.lib

Construct an interval [a, b] from two numeric bounds (order-independent).

interval.lib

A box, i.e., a list of intervals (typically one interval per variable).

interval.lib

Construct a box from a list of intervals, matching the number of intervals to the number of ring variables.

interval.lib

Return a box whose intervals are the elementwise centers of the input box.

interval.lib

Test whether two boxes have identical bounds in every component; returns 1 if equal, otherwise 0.

interval.lib

Return the n-th interval contained in a box (equivalent to B[n]).

interval.lib

Compute the intersection of one or more boxes; returns the intersected box or -1 if empty.

interval.lib

Test whether box A is strictly contained in the interior of box B (coordinate-wise).

interval.lib

Pretty-prints a box.

interval.lib

Set the n-th interval of a box to a given interval and return the updated box.

rootisolation.lib

Return a new box equal to B except with the i-th entry replaced by interval I.

interval.lib

Componentwise subtraction of two boxes (intervals per variable): returns B - C.

polybori.lib

PolyBoRi polynomial type corresponding to Singular's @code{poly}.

polybori.lib

Adds two bpoly Boolean polynomials and returns the resulting bpoly.

polybori.lib

Test whether two bpoly objects represent the same underlying boolpoly.

polybori.lib

Multiply two bpoly Boolean polynomials and return the product as a bpoly.

polybori.lib

Convert a Boolean polynomial (bpoly) into a Boolean ideal (bideal).

polybori.lib

Convert a Boolean polynomial (bpoly) to a Boolean set (bset) by extracting its underlying set.

polybori.lib

Convert a PolyBoRi boolean polynomial (bpoly) into a Singular poly.

polybori.lib

Return the underlying Python boolean polynomial object from a PolyBoRi bpoly.

plural.doc / bracket

Computes the Lie bracket [p,q]=pq-qp for two polynomials, with an optional third argument to compute a right-normed iterated bracket.

kskernel.lib

Internal recursive builder that appends a bracket expression to global string F using indices r and s.

examples.doc / Branches of space curve singularities

This example computes the number of branches of a quasihomogeneous isolated space curve singularity and illustrates pitfalls of primary decomposition. It cross-checks results using invariants and normalization.

tutor.tex / Branches of space curve singularities

This example computes the number of branches of a quasihomogeneous isolated space curve singularity and illustrates pitfalls in using primary decomposition.

reference.doc / break

Describes the break; statement, which exits the innermost for or while block. Includes a simple example and related references.

general.doc / Break points

Describes how to insert and use break points in a procedure in Singular, including interactive command input and an example session.

reference.doc / breakpoint

Sets a breakpoint at the start of a specified procedure or at a given line number, and can remove breakpoints for that procedure.

brnoeth.lib

Compute a basis of the Riemann–Roch space L(G) on a non-singular curve, returned as a list of ideals encoding functions H/Ho.

1 commands

This library provides procedures to compute vector space bases of Riemann–Roch spaces of divisors on projective curves.

polybori.lib

PolyBoRi ring type corresponding to Singular's @code{ring}.

13 commands

Implementation of the Brill–Noether algorithm for solving the Riemann–Roch problem, with applications to Algebraic Geometry codes.

polybori.lib

PolyBoRi set type corresponding to the @code{zdd} type used here.

polybori.lib

Convert a PolyBoRi boolean set (bset) to a boolean polynomial (bpoly).

polybori.lib

Convert a Boolean set (bset) to a zero-suppressed decision diagram (zdd).

dmodideal.lib

Compute Bernstein–Sato-type ideal(s) for f=F[1]..F[P] from Ann(F^s), returning a new ring containing the ideal BS.

redcgs.lib

Builds and returns a string assignment of the form T[[labels,3]]:=[L[3]].

redcgs.lib

Builds two branch-condition assignment strings (L[8]=0 and L[8]<>0) using labels from L[1].

redcgs.lib

Builds and returns a string assignment of the form T[[labels,12]]:=[...]; using L[1] and L[7].

redcgs.lib

Builds and returns an assignment string of the form T[[lab,4]]:=[...], with special handling when lab=[-1].

redcgs.lib

Builds and returns a string assignment T[[labels,6]]:=true/false based on list L.

redcgs.lib

Builds and returns a formatted assignment string for T at indices from L[1], inserting the contents of L[5].

crypto.lib

Sort a list in decreasing order using bubble sort.

math.doc / Buchberger algorithm

Describes improvements to the Buchberger algorithm for toric ideals, leveraging their special structure. Mentions strategies and references relevant to toric and more general ideals.

Building-Singular-from-source.md

Generic installation notes for building Singular from source, including development and old master branches, static and debug variants, testing, and coverage measurement with lcov.

integralbasis.lib

Builds factor-related lists (factors, orders, degrees, checks) from a list of class descriptors used in integralbasis.lib.

resgraph.lib

Return a writable filename: use given name if writable, else try in /tmp; error on path names or if /tmp is not writable.

platform.doc / Building Singular

Instructions for building and installing Singular using a standard UNIX-like workflow, with notes on configuration options and side-by-side installation.

integralbasis.lib

Build a polynomial in an extended ring by adjoining a den-th root of unity and truncating computations to degree d.

integralbasis.lib

Builds and returns the polynomial output (first entry) from buildPolyGroundXRoot, optionally switching to a given ring.

redcgs.lib

Builds a dichotomic discussion tree (first reduced CGS) for an ideal in K[a][x], optionally starting from given null/non-null conditions.

redcgs.lib

Write a buildtree output list to a file as a Maple-readable table.

tateProdCplxNegGrad.lib

Data structure representing a (Beilinson) bundle used in Beilinson-type constructions.

linalg.lib

Compute characteristic polynomial coefficients and busadjoint coefficient matrices for a square matrix A.

polybori.lib

Return the i-th variable of the Boolean ring corresponding to the current basering.

C-STYLEGUIDE.md

Legacy C/C++ style guide for Singular, covering file naming, structure, headers, formatting, naming conventions, comments, debugging aids, and C++ feature usage.

crypto.lib

Returns the sum of all elements in a list.

crypto.lib

Finds an exponent x such that primitive^x ≡ num1 (mod mod1) (returns 0 if none found in the searched range).

resbinomial.lib

Compute updated exponent data expI by adjusting the divisor D from (c,oldordI,Y,a,oldD) and reducing expJ accordingly.

gfan.lib

Return a canonical (unique) representation of a given cone.

control.lib

Canonize modules in a list by replacing each module entry with its reduced minimal Groebner basis (std), preserving other entries.

homolog.lib

Compute kernel (two representations) and cokernel of the canonical map M -> Ext^c(Ext^c(M,R),R) from a presentation.

redcgs.lib

Convert CGS output (mrcgs/rcgs/crcgs) into a simpler list where each segment is given as V(I1) V(I2) with radical ideals.

hyperel.lib

Add two semireduced divisors on a hyperelliptic curve via Cantor's composition algorithm.

hyperel.lib

Compute the reduced multiple mD of a (semi)reduced divisor D on the hyperelliptic curve y^2+h(x)y=f(x) using repeated doubling.

hyperel.lib

Reduce a semireduced divisor on a hyperelliptic curve using Cantor's reduction algorithm.

redcgs.lib

Builds a canonical tree list from a list of select cases by expanding each case via GCS and sorting labels.

redcgs.lib

Write a CGS tree/list (from mrcgs/rcgs/crcgs) to a file as a Maple table with the given table variable name.

cisimplicial.lib

Return the cardinality of Z^m/ZA for columns of an integer matrix A (0 if infinite), optionally using only first n columns.

general.doc / Category string

A category string is part of a library header. It defines the library category and is reserved for sorting libraries into categories.

nets.lib

Horizontally concatenate two Nets, aligning them at the top row.

tropicalNewton.lib

Convert the first row of a matrix into a strictly positive equivalent weight intvec (constant coefficient case).

ncdecomp.lib

Decompose a module into weight modules by central characters with respect to given central generators, if a finite decomposition exists.

central.lib

Compute generators of the center of an nc algebra, up to a degree bound D and/or until at least N elements are found.

resolve.lib

Compute the center ideal of the blow-up of J (optionally relative to W and exceptional divisors E) for the resolution algorithm.

resolve.lib

Compute the center ideal for the next blow-up in the Bierstone–Milman resolution algorithm from a BM presentation object.

resolve.lib

Compute the center ideal for the next blow-up in the Bravo–Encinas–Villamayor resolution algorithm for a given basic object.

central.lib

Compute subalgebra generators of the center up to a given degree bound D (optionally stop after finding N elements).

resolve.lib

Auxiliary BM-algorithm routine returning center data (as a list) computed from BM state and a center ideal.

central.lib

Compute an ideal generated by a vector space basis of center elements up to degree D.

plulibs.doc / central_lib

A library for computing elements of the center and centralizers of sets of elements in GR-algebras.

20 commands

A library for computing elements of the center and centralizers of sets of elements in GR-algebras.

finitediff.lib

Return a predefined first-order central finite-difference approximation (as a vector) for two derivative names with respect to a variable.

finitediff.lib

Return a predefined second-order central finite-difference approximation relating two derivative names in a given variable.

central.lib

Compute generators of the centralizer of a polynomial/ideal up to a degree bound D and/or until at least N elements are found.

central.lib

Compute a set of subalgebra generators for centralizer(F), truncated by degree D and optionally by count N.

central.lib

Compute an ideal generating a vector space basis of the centralizer of F up to degree d.

central.lib

Compute a vector space basis (as an ideal) of the centralizer of F within the vector space generated by V.

ncdecomp.lib

Compute the central quotient M:G for a module M by a central ideal G.

ncdecomp.lib

Compute the central saturation M:T^infinity by repeatedly applying CentralQuot with a central ideal T.

tropicalEllipticCovers.lib

Compute the coefficient of a given monomial g in a polynomial f.

compregb.lib

Compute a comprehensive Groebner system for an ideal with respect to given variables and parameters.

grobcov.lib

Compute a disjoint, reduced Comprehensive Groebner System (CGS) for an ideal in Q[a][x], with optional algorithm/output restrictions.

teachstd.lib

Test whether the chain criterion applies to f,g,h (polynomials or module elements), returning 1 if applicable, else 0.

chern.lib

Compute the positive-degree terms (optionally up to degree N) of the Chern character from a list of Chern classes.

chern.lib

Compute Chern classes from the positive-degree terms of the Chern character (inverse to chAll()).

start.doc / Change of rings

Explains how to compute the local Milnor number by switching to a ring with local ordering, and how to work with the Hessian matrix, its determinant, and minors in Singular.

tutor.tex / Change of rings

Explains how to switch to a local ring ordering in Singular to compute the local Milnor number and related invariants, and how to work with the Hessian matrix and its minors.

ring.lib

Return a copy of a ring with its characteristic (ring_list(r)[1]) replaced by the given list.

VecField.lib

Transform a vector field to new coordinates via an algebra morphism (inverting the map up to the field’s precision).

normal.lib

Change the denominator of an A-module 1/c1U1 over A=R/I to 1/c2U2, returning U2 (assumes such U2 exists).

ring.lib

Create a new ring (or qring) from an existing one by replacing its monomial ordering with a given ordering list.

ring.lib

Change the monomial ordering of a ring to a specified simple ordering.

resbinomial.lib

Rename basering variables to x(1..n) (if needed) and return the new ring together with a flag.

ring.lib

Create a new ring from a given ring/qring by changing its variable names according to a string specification.

reference.doc / char

Returns the characteristic of the coefficient field of a ring. The result is an integer.

reference.doc / char_series

Describes the Singular command char_series, which returns a matrix whose rows represent the irreducible characteristic series of an ideal with respect to the current variable ordering.

math.doc / Characteristic sets

Defines characteristic (triangular) sets and irreducible triangular systems in a polynomial ring with lexicographic order, and states the decomposition theorem into an irreducible characteristic series.

curvepar.lib

Compute the characteristic exponents of the branches of a plane curve V(i).

schubert.lib

Call signature for characters from schubert.lib (no further information available).

alexpoly.lib

Compute the conductor (an int) of an irreducible plane curve singularity from its characteristic exponents.

alexpoly.lib

Compute a minimal set of semigroup generators of an irreducible plane curve singularity from its characteristic exponents.

alexpoly.lib

Compute the intersection multiplicity matrix of plane curve branches from their contact matrix and characteristic exponents.

alexpoly.lib

Convert characteristic exponents of an irreducible plane curve singularity into its multiplicity sequence.

alexpoly.lib

Constructs a plane curve polynomial with given characteristic exponents from parametrization coefficients.

dmodapp.lib

Compute characteristic variety of a Weyl-algebra ideal, plus its singular locus and a primary decomposition, returned via a commutative ring.

hnoether.lib

Compute the characteristic polynomial of a bivariate polynomial whose Newton polygon is a single line.

linalg.lib

Compute the characteristic polynomial det(Ev - A) of a square matrix (default v is the last ring variable).

linalg.lib

Compute the characteristic polynomial det(Ev - A) of a square matrix A, using variable v (default: last basering variable).

reference.doc / charstr

Describes the coefficient field of a ring as a string. For coefficient-type tests, prefer the routines in ring.lib because the string representation may change.

scheme.lib

An affine chart of the form Spec R/I.

scheme.lib

A morphism from an open subset of a chart to another chart, given by rational functions on coordinates.

scheme.lib

Compute the composition of two compatible chartmaps (phi after psi).

scheme.lib

Compute the ideal defining the inverse image under a chartmap of the closed subscheme given by an ideal.

scheme.lib

Convert a chartmap into the corresponding morphism.

scheme.lib

Convert a chart into the corresponding affine scheme.

dmodapp.lib

Compute a commutative output ring containing the ideal charVar whose zero set is the characteristic variety of I.

chern.lib

Compute Chern classes of the dual vector bundle by negating the odd-indexed Chern classes.

decomp.lib

Returns the (generalized) nth Chebyshev polynomial of the first kind in the first variable of the basering.

dmod.lib

Test whether a given qs in K[s] divides the global Bernstein–Sato polynomial of f using Ann(f^s) in D[s].

pfd.lib

Test whether a rational function f/g equals a given partial fraction decomposition (exactly or via randomized evaluation).

pfd.lib

Check correctness of pfdMat output (indexed decompositions) against the input and denominator factors, optionally via random evaluation and/or parallelism.

dmod.lib

Test whether -alpha is a root of the global Bernstein–Sato polynomial of f (and optionally return its multiplicity).

dmod.lib

Test whether -alpha is a root of the global Bernstein–Sato polynomial of f, given the annihilator ideal of f^s in D[s].

dmod.lib

Compute the multiplicity of -alpha as a root of the Bernstein–Sato polynomial of f from a known annihilator ideal Ann(f^s) in D[s].

chern.lib

Compute Chern classes (as polynomials) from Segre classes given by a list of polynomials.

61 commands

## Symbolic Computations with Chern Classes

chern.lib

Compute the polynomial in t corresponding to a Chern character from r and a list of components ch.

schubert.lib

Returns the i-th Chern class of a sheaf as a polynomial.

chern.lib

Compute the Chern polynomial in t from a list of Chern class polynomials.

chern.lib

Return the Chern roots (sign-changed) of the dual vector bundle for a given list of Chern roots.

chern.lib

Compute the Chern roots of Hom(E,F) from the Chern roots of vector bundles E (a) and F (b).

chern.lib

Compute Chern roots of the tensor product from two lists of Chern roots (polynomials).

chern.lib

Compute Chern roots of the Whitney (direct) sum of two bundles given by their Chern-root lists.

chern.lib

Compute the Chern roots of the m-th symmetric power of a vector bundle with Chern roots given by l.

chern.lib

Compute the Chern roots (as polynomials) of the m-th exterior power from a list of Chern roots.

chern.lib

Compute the highest relevant term of the Chern character from a list of Chern class polynomials.

chern.lib

Compute (up to degree N) the Chern classes of Hom(E,F) from ranks and Chern classes of E and F.

crypto.lib

Computes x satisfying x = T[i] mod L[i] for all i (Chinese remainder theorem).

reference.doc / chinrem

Applies the Chinese remainder theorem to elements of the first argument with respect to moduli given in the second. Supports several Singular types and can be applied recursively to nested lists.

nfmodstd.lib

Chinese remainder construction for poly/ideal/module data with respect to a list of coprime polynomial moduli.

hyperel.lib

Chinese remainder theorem for polynomials: compute poly x with x ≡ b[i] (mod moduli[i]) for all i.

chern.lib

Compute the Chern-number monomials (partitions of N) from a list of Chern classes on an N-dimensional complex manifold.

chern.lib

Compute the Chern numbers of a vector bundle from its Chern classes on a complex manifold of dimension r.

chern.lib

Compute the Chern numbers of the projective space P_n.

schubert.lib

Construct the Chow ring of an abstract variety as a quotient ring of its base ring by its relations.

chern.lib

Compute the Chern classes of the product (tensor product) of two vector bundles from their ranks and Chern class lists.

chern.lib

Compute the Chern classes of the product of two vector bundles from their Chern class lists (via elimination).

chern.lib

Compute the Chern classes (as a list) of the product of two vector bundles from their Chern class lists.

chern.lib

Compute the total Chern class of the product of two vector bundles from their ranks and Chern class lists (Lascoux formula).

chern.lib

Compute the list of Chern classes of the product of two vector bundles from their Chern class lists (Manivel formula).

chern.lib

Compute the total Chern class of the product of two vector bundles from their ranks and Chern classes.

chern.lib

Compute the coefficients of the Chern character of the tangent bundle on complex projective space of dimension n.

chern.lib

Compute Chern class polynomials of the direct sum of two vector bundles given their Chern class lists.

chern.lib

Compute rank and Chern classes of the k-th symmetric power of a rank-r vector bundle from its Chern classes.

chern.lib

Compute Chern classes of the 2nd symmetric power of a rank-r vector bundle from its Chern classes.

chern.lib

Compute the total Chern class of the second symmetric power of a rank r vector bundle from its Chern classes.

chern.lib

Compute rank and Chern classes of the k-th exterior power of a rank r vector bundle from its Chern classes (optionally only selected indices).

chern.lib

Compute Chern classes of the second exterior power of a rank r vector bundle from its Chern classes.

chern.lib

Compute the total Chern class of the second exterior power of a rank r vector bundle using Lascoux's formula.

3 commands

A library to determine whether the toric ideal of an affine monomial curve is a complete intersection, without explicitly computing a generating set for the ideal.

6 commands

A library for determining whether a simplicial toric ideal is a complete intersection, without explicitly computing a generating set of the ideal. The procedures are based on:

examples.doc / Classification of hypersurface singularities

Describes how to classify isolated hypersurface singularities up to right equivalence using classify.lib, based on Arnold’s algorithm and an initial guess from the Milnor algebra’s Hilbert polynomial.

tutor.tex / Classification of hypersurface singularities

Describes how to classify isolated hypersurface singularities up to right equivalence using classify from classify.lib, based on Arnold’s algorithm and an initial guess via the Milnor algebra’s Hilbert polynomial.

classify.lib

Computes the normal form and singularity type of a polynomial (Arnold et al. classification) and returns the normal form as a poly.

9 commands

A library for classifying simple singularities with respect to A-equivalence in characteristic 0.

classify.lib

Initialize (and if needed create) the classify.lib DBM database NFlist and read its VERSION.

22 commands

A library for classifying isolated hypersurface singularities with respect to right equivalence, based on the determinator of singularities by V.I. Arnold.

2 commands

We classify isolated singularities of corank <=2 and modality <=2 with respect to right-equivalence over the complex numbers, according to Arnold's list. We determine the type and, in the case of positive modality, the parameter.

classifyceq.lib

Classify a hypersurface singularity (Greuel–Kröning) and return a string with its type and Tjurina number.

1 commands

A library for classifying simple hypersurface singularities with respect to contact equivalence in characteristic \(p > 0\).

2 commands

A library for classifying isolated complete intersection singularities over a base field of characteristic 0, and for computing the Weierstrass semigroup of a space curve.

classifyci.lib

Classify an ideal with two generators as an isolated simple complete intersection singularity (char 0), else return a message.

8 commands

A library for computing a standard basis of the tangent space at the orbit of an algebraic group action.

classifyMapGerms.lib

Computes the Rieger normal form of a simple map germ given by a 2-generated ideal in 2 variables; returns (0,0) if not simple.

classifyMapGerms.lib

Classify a 2-generator ideal defining a C^2->C^2 map germ; returns its Rieger simple normal form or ideal(0,0) if not simple.

classifyMapGerms.lib

Classify a 2-generator map germ C^2->C^2 by returning its Rieger normal form ideal, or ideal(0,0) if not unimodal.

chern.lib

Builds a polynomial in t with coefficients taken from the entries of l (sum_{i=1..n} l[i]t^i).

classify_aeq.lib

Compute the normal form of a parametrized space curve ideal if it is simple; otherwise return -1.

redcgs.lib

Map ideal to ring @RP, clear denominators of non-empty generators, and map the resulting ideal back to the original basering.

tropical.lib

Remove temporary tropical.lib LaTeX/PS-related files from /tmp matching tropicalcurve and tropicalnewtonsubdivision.

resbinomial.lib

Clean the units in a monomial.

stratify.lib

Join entries of a strata-like list that share the same leading intvec into a single entry, recording alternative open-set data.

reference.doc / cleardenom

Describes cleardenom, which clears denominators in coefficients of a polynomial or vector by multiplying by a suitable constant and then dividing by the content.

reference.doc / close

Closes a link. The example shows starting and shutting down a Singular server via an ssi:tcp link.

brnoeth.lib

Return the distinct affine closed points of V(I) as a list of prime ideals (Groebner bases).

ssi.lib

Closes a link to an SSI file.

latex.lib

Write the closing command for a TeX/LaTeX document into .tex (optionally selecting the TeX dialect).

normal.lib

Compute a fraction representation f = num/den using the injections (phi) along a primeClosure chain; stores result in global list fraction.

normal.lib

Compute generator preimages along a chain of rings R(1)->...->R(n) in the current Rees algebra basering.

reesclos.lib

Compute generators in k[x,t] for the integral closures of powers of an ideal from generators of the closure of R[It].

reesclos.lib

Compute images in k[x,t] of generators of the integral closure of a Rees algebra presentation, returned as numerators plus a universal denominator.

sheafcoh.lib

Compute the Castelnuovo–Mumford regularity of coker(M) for a graded module with an admissible degree vector.

spcurve.lib

Return the Cohen–Macaulay type of an ideal of codimension 2, or -1 if it is not Cohen–Macaulay of codimension 2.

crypto.lib

RSA-encodes a lowercase-and-space string and returns the ciphertext as a string.

math.doc / Codes and the decoding problem

Introduces linear and cyclic codes, key distance notions, and the decoding problem. Describes generic vs. online decoding via solving systems and compares their computational effort.

classifyMapGerms.lib

Compute the K-vector space dimension of A^r/(M+N+maxideal(bound)A^r), optionally correcting the bound via determinacy.

sing.lib

Compute the vector space dimension of id1/id2 if finite (with containment checks), otherwise return -1 (infinite) or -2 (not contained).

gfan.lib

Return the codimension of a cone, fan, or polytope.

classifyMapGerms.lib

Compute a bound for the A-determinacy of the map germ defined by an ideal and its (extended) codimension.

mondromy.lib

Compute the codimension of the subspace generated by V inside the ambient vector space of dimension pcvdim(0,N).

reference.doc / coef

Describes the Singular command coef and its syntax for polynomials, ideals, and vectors, including returned matrices and behavior compared to coeffs. Includes examples.

arnold.lib

Return the coefficient of a given monomial m in a polynomial f.

general.doc / Coefficient rings

Describes coefficient rings in Singular that are not fields and introduces p-adic integers as limits of finite rings.

aksaka.lib

Reduce the coefficients of a univariate polynomial (in var(1)) modulo a given bigint.

reference.doc / coeffs

Documents the coeffs command in Singular, including syntax, return types, and examples for extracting coefficients from polynomials, ideals, vectors, modules, and rings.

interval.lib

Return the cofactor of an interval matrix A at position (i,j).

nchomolog.lib

Compute the matrix of the induced homomorphism Hom(R^s,F1) -> Hom(R^s,F2) from a map given by matrix M.

tateProdCplxNegGrad.lib

Compute a hash table of dimensions dim(H^i(F(c))) for twists c in the intvec range [low, high] for the sheaf coker(M).

tateProdCplxNegGrad.lib

Computes and exports the cohomology matrix for the sheaf associated to coker(M) over a product of two projective spaces.

tateProdCplxNegGrad.lib

Compute the cohomology matrix (exported as cohomologymat) for a Tate-resolution multigradedcomplex in a given Z^2 degree range.

modules.lib

Compute the cokernel module of a matrix of relations.

reszeta.lib

Identify and match exceptional divisors across the resolution charts and return identification tables.

control.lib

Compute the column rank of a module/matrix using the Bareiss algorithm.

matrix.lib

Compute the column-reduced form of a matrix, optionally also returning the transformation matrix.

3 commands

Some basic combinatorics utilities.

general.doc / Command line options

Describes the startup syntax for Singular/ESingular and documents all supported command-line options, including debugging, startup control, timing, randomness, ssi links, and Emacs integration.

types.doc / Commands for user defined types

Describes how to install procedures so that kernel commands can operate on user-defined types. Includes the list of supported kernel commands and an example.

dmodloc.lib

Return the basering with any non-commutative relations removed.

examples.doc / Commutative Algebra

Index of examples and topics in commutative algebra. Links point to sections on standard computations such as saturation, elimination, resolutions, and decomposition.

tutor.tex / Commutative Algebra

Table of contents for the Commutative Algebra section, listing the available topics and their links.

redcgs.lib

Compute a list via recursion on n with base case list(0:p), merging unique elements from intermediate results.

redcgs.lib

Return list of intvecs obtained by incrementing each component of l by 1 (special-cases size 0/1).

modules.lib

Compare two Matrix objects for equality; returns 1 if equal, otherwise 0.

modules.lib

Compare two modules up to isomorphism; returns 1 if isomorphic, otherwise 0 (also used by M==N).

tasks.lib

Compare two tasks for equality of metadata (id, command, state, linkID), ignoring arguments and results.

modules.lib

Compare two vectors taking into account the relations of their module.

bimodules.lib

Decompose a polynomial in an enveloping algebra into an ideal whose generators group terms with the same right tensor component.

cimonom.lib

Test whether the toric ideal I(d) is a complete intersection (returns 1 if yes, 0 otherwise).

sets.lib

Compute the complement of set S in set M (requires S ⊆ M).

chern.lib

Compute complete homogeneous symmetric polynomials up to degree N in terms of elementary symmetric polynomials given in c.

scheme.lib

Constructs a cover/isomorphic scheme whose charts are complete intersections, returning the scheme and mutually inverse morphisms.

ainvar.lib

Compute the complete SAGBI reduction of p w.r.t. dom by repeatedly applying reduction until it stabilizes.

ainvar.lib

Compute the complete SAGBI reduction of p with respect to dom by repeatedly applying reduction (optionally with q and/or n).

classify2.lib

Classify a plane curve singularity (modality<=2, corank<=2) and return a normal form equation as a NormalFormEquation object.

classify.lib

Classify a complex singularity given by a polynomial and return its type as a string.

classify2.lib

Classify a bivariate curve singularity over C (modality, corank <= 2), returning its complex type and modality.

finitediff.lib

Return the formal complex value (squared modulus) of a polynomial, treating var(1) as the imaginary unit.

resolve.lib

Internal predicate: tests whether J meets some nontrivial ideal in the list E (based on deg(std(...)[1]) != 0).

ehv.lib

Compute a P-primary component Q of an ideal I using an EHV-based strategy.

decomp.lib

Compute iterated polynomial composition f1(f2(...(fn))) from polynomials or from an ideal listing the entries.

4 commands

Experimental implementation of comprehensive Gröbner systems.

matrix.lib

Delete zero columns (or zero generators/elements) from a matrix/ideal/module/intmat/intvec and return the same type.

schubert.lib

Calls procedure computation from schubert.lib.

examples.doc / Computation of Ext

Shows how to compute \mathrm{Ext}^n via a resolution, dualization, and homology, with examples and references to homolog.lib. Includes procedures for deformation- and obstruction-related Ext-modules.

tutor.tex / Computation of Ext

This section explains how to compute \mathrm{Ext}^n via resolutions, transposition, syzygies, and quotients, and demonstrates more general procedures from homolog.lib with examples.

deform.lib

Computes Ext^1, Ext^2 and auxiliary matrices/modules (for mod_versal) from a presentation matrix.

gitfan.lib

Compute the orbits of afaces in AF under the group action given by the permutation list G.

ellipticcovers.lib

Return the constant coefficient of the Laurent series of a rational function f in the variable x.

ellipticcovers.lib

Compute Gromov–Witten invariants for a Feynman graph over a range of partitions, returning a list with zeros outside the range.

resbinomial.lib

Compute the least common multiple of the denominators of the given E-orders (numbers) for one chart.

reszeta.lib

Compute multiplicities of each divisor in the total transform under a resolution.

gitfan.lib

Compute minimal hash vectors for all neighboring cones of the cone identified by lambdaHash.

gitfan.lib

Compute minimal (under a given action) hash representatives of the neighbour cones of a cone identified by lambdaHash.

reszeta.lib

Compute multiplicities of exceptional divisors in the pullback of a volume form (returns an intvec indexed by divisors).

examples.doc / Computing Groebner and Standard Bases

Lists the Groebner/standard basis algorithms available as optional arguments for various ideal and module operations in Singular, and the functions that support this choice.

tutor.tex / Computing Groebner and Standard Bases

Lists the available Gröbner/standard basis algorithms and the functions that can select among them via an optional string argument.

matrix.lib

Concatenate multiple matrices side-by-side into one matrix with row count equal to the maximum among inputs.

finvar.lib

Concatenate two integer matrices horizontally (same number of rows).

tateProdCplxNegGrad.lib

Concatenate two integer matrices horizontally (same row count), returning the combined intmat.

kernel

In order to use convex objects in Singular, Singular uses gfanlib, a C++ library for convex geometry by Anders N. Jensen. Please check the readme file for installation instructions. The library gfan_lib provides this C++ library and an interface to it.

cones.doc / cone

Notes the build requirement for using convex objects in Singular and states that this version does not support cone.

types.doc / cone

Introduces convex rational polyhedral cones in Singular via gfanlib, including their basic definitions and the two main representations used in Singular.

gfan.lib

Compute the link of a cone around a given vector or matrix input.

cones.doc

gfan.lib

Construct a cone from linear inequalities IEx>=0 and optional equations Ex=0, removing redundancies and using optional assumption flags.

gfan.lib

Construct a cone from half-lines given by rows of HL and optionally lines/lineality space given by rows of L, with optional assumptions via flags.

tropical.lib

Compute the conic through five points, its tangents and tropicalisations; optionally also for the dual conic.

grobcov.lib

Constructs the canonical level closures of a constructible set (given as locally closed pieces) and its complement.

resbinomial.lib

Constructs the ideal defining the composed blow-up map to a chart by iterated substitutions according to a blow-up history.

resbinomial.lib

Constructs the list of exceptional divisor ideals for a chart from a blow-up history vector.

resbinomial.lib

Constructs the ideal defining the last blow-up by applying substitutions determined from a blow-up history.

JMBTest.lib

Builds and orders intermediate data from G and c, then multiplies/reduces to return a list result.

curvepar.lib

Compute the contact matrix of the branches of the plane curve V(I).

redcgs.lib

Tests whether all generators of Pu reduce to 0 modulo Pv (i.e., Pu is contained in Pv).

zeroset.lib

Test whether f is contained in data (membership predicate).

gfan.lib

Test whether cone d is a face of cone c.

gfan.lib

Test whether a fan (collection) contains a given cone; returns 1 if contained, 0 otherwise.

gfan.lib

Test whether a cone or a point (given as intvec/bigintmat) is contained in the support of a cone.

gfan.lib

Test whether a cone contains a vector with only positive entries in its relative interior.

gfan.lib

Test whether a cone contains a given point in its relative interior (returns 1 if yes, else 0).

polylib.lib

Compute the content (gcd of coefficients) of a polynomial or vector.

math.doc / Conti and Traverso

Describes the Conti–Traverso algorithm for computing the toric ideal I_A using an extended matrix and elimination. Notes practical performance issues and relevance to integer programming.

reference.doc / continue

Describes the continue; statement, which skips the remainder of the innermost loop body and continues execution at the start of the loop block.

reference.doc / contract

Describes the contract command, which contracts elements of one ideal by elements of another and returns a matrix. Includes the monomial definition of contraction and an example.

nchomolog.lib

Compute the matrix of the induced contravariant map Hom(F2,R^s) -> Hom(F1,R^s) from a matrix M: F1->F2.

schubert.lib

Compute the contribution bundle factor for a fixed-point component (graph) in a moduli space of stable maps.

control.lib

Compute a list of controllability-related properties of the system/behavior represented by a module (system matrix) R.

reference.doc / Control structures

Overview of Singular control structures and the block syntax used in procedures and conditional/loop statements.

equising.lib

Compute control matrices (multiplicities, branch counts, tangent splitting, degree bounds) from a multsequence/hnexpansion list.

17 commands

Algebraic analysis tools for System and Control Theory.

control.lib

Compute a list of controllability-related properties for a behavior given by a full-row-rank system matrix/module R using dimension calculations.

control.lib

Load a predefined control example by name (or list examples with "show"), initializing a ring (and predefined module R).

deRham.lib

Constructs and returns a noncommutative algebra (via nc_algebra) from the current basering with a specific commutation matrix.

resbinomial.lib

Reconstruct an ideal from coefficient list C and exponent list L (e.g. output of data), using n and a variable flag list.

olga.lib

Convert a given fraction to a right fraction representation for the specified localization.

olga.lib

Convert a right fraction representation to a left fraction representation for a given localization.

gfan.lib

Computes the convex hull of two cones/polytopes, returning a cone for cone+cone and a polytope otherwise.

gfan.lib

Compute the convex intersection of two cones/polytopes or of a list of cones/polytopes.

dmod.lib

Convert a ringlist by replacing all 'p' orderings with corresponding 's' orderings (e.g., dp -> ds).

decodegb.lib

Convert each element of a list using list2vec and return the modified list.

math.doc / Cooper philosophy

Overview of Cooper’s approach for decoding cyclic codes via CRHT syndrome ideals and general error-locator polynomials, and an adaptation for computing minimum distance.

rootsmr.lib

Compute the coordinate list of f mod I with respect to the ordered monomial basis B of R/I.

tasks.lib

Create and return an independent copy of a task (cannot copy a task whose state is 'started').

classify.lib

Compute the corank of the Hessian matrix of a polynomial f.

pointid.lib

Compute the corner monomials (leading monomials of a compatible ideal) of a monomial order ideal N.

crypto.lib

Counts points on the elliptic curve y^2=x^3+ax+b over Z/N using a trivial Jacobi-sum approach.

tropicalEllipticCovers.lib

Compute the cover multiplicities (one per cover) for an elliptic curve cover with source graph G, for a given vertex ordering.

tropicalEllipticCovers.lib

Compute, for each elliptic cover encoded by (G,aa,O,t[,l]), a list of edge-tuples (a_k,N_k,D_k) from propagator data.

chern.lib

Compute the positive-degree coefficients of the total Chern class of the tangent bundle on complex projective space of dimension n.

redcgs.lib

Compute the Canonical Reduced CGS (CRCGS) of an ideal, optionally restricting parameters by given null/non-null conditions.

standard.lib

Construct a ring from the four components of ring_list(R), accepting list/int/string shortcuts and optional quotient ideal and "no_minpoly".

reference.doc / create_ring

Reference entry for the create_ring topic in Singular.

makedbm.lib

Create/populate a DBM database link "NFlist" with predefined entries (e.g., A[k], D[k], E[6k], ...) and their invariants.

SingularityDBM.lib

Creates the DBM database files Singularitylist.dir and Singularitylist.pag for Arnol'd singularities up to corank 2.

resolve.lib

Create a basic object (BO) list from an ideal J with optional ideals W and exceptional divisors list E.

multigrading.lib

Create a multigraded ring homomorphism from a ring map (via generator images) and a grading-group homomorphism.

multigrading.lib

Create the group (S+L)/L from generator matrix S and relation matrix L.

resbinomial.lib

Builds a list of 2-element sublists pairing L1[i] with L2[i], dropping entries where L1[i] is 0.

tateProdCplxNegGrad.lib

Convert a list (shift + multigraded differentials) into a multigradedcomplex object.

finitediff.lib

Create a Perl filter script qepcadfilter.pl in the current directory, removing any existing file first.

multigrading.lib

Create the quotient group (I+L)/L, where I is the identity matrix of size nrows(L).

tasks.lib

Create a task for the given command and argument list in state 'created'.

multigrading.lib

Create the torsion-free (free) subgroup generated by S inside the free abelian group Z^n with n = nrows(S).

grobcov.lib

Compute the canonical C-representation [P,Q] (radical ideals) of the locally closed set V(N) V(M).

kernel

Variables of type cring represent the ring of coefficients (see number)

types.doc / cring

Documents the Singular type cring, which represents the ring of coefficients, and links to its declarations, expressions, operations, and related functions.

types.doc / cring declarations

Describes the syntax and purpose of cring declarations in Singular, including default behavior and a short example.

types.doc / cring expressions

Defines cring expressions in Singular and lists the valid forms, with a short example.

types.doc / cring operations

Documents the division operator for creating a residue class ring in a coefficient ring context.

types.doc / cring related functions

Overview of functions related to the type cring, including cross products and numeric/rational function variants.

redcgs.lib

Tests a pair (i,j) against a criterion using lcm of leading monomials and a list of excluded pairs, returning 0/1.

examples.doc / Critical points

Explains how to compute the number of critical and singular points of a polynomial using a different monomial ordering, and illustrates the method with an example in Singular.

tutor.tex / Critical points

Explains how to compute the number of critical and singular points of a polynomial in affine space by switching to ordering dp and working in K[x_1,\ldots,x_n]. Includes an example in Singular using imap, jacob, and vdim.

reference.doc / crossprod

Documents the Singular function crossprod, which computes the cross product of several objects of type cring.

66 commands

The library contains procedures for computing discrete logarithms, performing primality tests, and factorization (including methods using elliptic curves).

fpalgebras.lib

Create the Letterplace ring for the crystallographic group C2MM and export ideal I with the defining relations.

fpalgebras.lib

Create a letterplace ring for a crystallographic CM group, exporting ideal I with the defining relations.

fpalgebras.lib

Create the Letterplace ring for the crystallographic group p1 and export ideal I with its defining relations.

fpalgebras.lib

Constructs the Letterplace ring for the crystallographic p2 group and exports the defining relations ideal I.

fpalgebras.lib

Construct the Letterplace ring for the crystallographic group p2gg with relations exported as ideal I.

fpalgebras.lib

Create the Letterplace ring for the crystallographic group p2mm with an exported ideal I of defining relations.

fpalgebras.lib

Construct the Letterplace ring for the crystallographic p3 group; exports ideal I of defining relations.

fpalgebras.lib

Constructs a Letterplace free algebra ring for the crystallographic group p31m and exports the defining relations ideal I.

fpalgebras.lib

Create the Letterplace ring for the crystallographic group p3m1 and export the defining relations as ideal I.

fpalgebras.lib

Create a Letterplace free algebra ring for the crystallographic p4 group; exports ideal I with defining relations.

fpalgebras.lib

Constructs the Letterplace ring for the crystallographic group p4gm and exports the defining relations ideal I.

fpalgebras.lib

Constructs the Letterplace ring for the crystallographic group p4mm and exports the ideal I of defining relations.

fpalgebras.lib

Constructs the Letterplace ring for the crystallographic p6 group (relations exported as ideal I).

fpalgebras.lib

Create the Letterplace ring for the crystallographic p6mm group; exports ideal I of defining relations.

fpalgebras.lib

Create a Letterplace ring for the crystallographic group pg with presentation relations exported as ideal I.

fpalgebras.lib

Create a letterplace/free-algebra ring for the crystallographic pm group presentation and export the relation ideal I.

chern.lib

Compute the Chern–Schwartz–MacPherson classes of the projective variety defined by a homogeneous ideal.

redcgs.lib

Builds and returns a list of assignment strings and a basis-like object derived from tree(lab, L).

redcgs.lib

Recursively traverses a tree structure from L and writes generated assignment strings to a write() target.

homolog.lib

Compute (symmetrized) cup-product Ext^1(M',M') x Ext^1(M',M') -> Ext^2(M',M') for M' = coker(matrix(M)).

homolog.lib

Compute the cup-product map Ext^p(M',N') x Ext^q(N',P') -> Ext^(p+q)(M',P') and return its matrix (or bases list).

curveInv.lib

Compute the colength of derivations of R/I (summing contributions over branches; optional precomputed normalization list).

curveInv.lib

Compute the multiplicity of the conductor of a curve ideal, optionally reusing a precomputed normalization.

curveInv.lib

Compute the Deligne number e = 3delta - m of the curve R/I, optionally reusing precomputed normalization data.

curveInv.lib

Compute the delta invariant of the 1-dimensional ring R/I; returns -1 if the delta invariant is infinite.

4 commands

This library provides a collection of procedures for computing invariants of curve singularities. The invariants that can be computed are:

10 commands

Library curvepar.lib provides procedures related to the resolution of singularities of space curves and the computation of associated invariants such as parametrizations and the Weierstrass semigroup.

curvepar.lib

Compute parametrizations of the algebraic branches of a curve ideal at the singular point 0.

curvepar.lib

Compute an embedded resolution of the curve V(I), returning a new ring containing the chart list Resolve.

general.doc / Customization of the Emacs interface

Describes how to use Emacs customization to configure the SINGULAR Emacs interface, including where settings are stored and which customization groups are available.

2 commands

This library provides customized standard basis algorithms to improve the performance of other algorithms.

polymake.lib

Compute (as a string) the cycle length of the corresponding cycle in the dual tropical curve for a polygon boundary around an interior point.

polymake.lib

Return boundary lattice points (clockwise) of the convex polygon spanned by points connected to points[pt] via triangulation.

polylib.lib

Constructs the ideal of cyclic n-roots in the first n variables of the current basering.

examples.doc / Cyclic roots

Defines a procedure that constructs the cyclic roots ideal over the current basering and computes a standard basis along with basic numerical invariants.

tutor.tex / Cyclic roots

Defines a procedure to generate the ideal of cyclic roots over the current basering and computes a standard basis together with basic numerical invariants.

finvar.lib

Return the i-th cyclotomic polynomial as a poly in the first variable of the current ring.

hyperel.lib

Compute k-term local expansion coefficients of y around point P on the hyperelliptic curve y^2+h(x)y=f(x).

resbinomial.lib

Constructs lists of coefficients and exponent vectors for the first k generators of an ideal.

types.doc

This chapter describes all Singular data types in alphabetical order, including declaration syntax and how to build expressions of each type.

pyobject.doc

plural.doc / Data types BR_PLURAL_BR

Explains all data types of Plural in alphabetical order, including declaration syntax and how to build expressions. Defines the term "expression list" and links to general declaration syntax.

standard.lib

Return the current date and time as a string.

reference.doc / datetime

Reference entry for the Singular keyword or function datetime. This section currently contains only the heading.

types.doc / DBM links

DBM links provide access to data stored in a database as key/value string pairs, supporting reading, writing, deletion, and scanning of entries.

makedbm.lib

Reads the next key from a DBM link and returns it together with its associated value.

SingularityDBM.lib

Read the next entry from a DBM database referenced by l.

makedbm.lib

Read and print all key/value entries from a DBM database link.

SingularityDBM.lib

Read all entries from a DBM database referenced by l.

reference.doc / dbprint

Documents the dbprint command, which conditionally applies print to expressions based on a debug flag or on printlevel and voice. Includes syntax variants, notes, and an example.

classify.lib

Print trace/debug messages when the given level is enabled by the global @DeBug setting.

general.doc / Debugging tools

Overview of debugging aids in Singular. Includes hints for escaping from a stuck user-defined procedure and links to related debugging features.

mprimdec.lib

Compute the minimal primary decomposition of a zero-dimensional submodule in a 1-variable ring, optionally skipping components containing a given module.

redcgs.lib

Compute and reduce an S-polynomial of g1,g2 w.r.t. N1,N2 and decide a resulting ideal based on reductions and nonnull tests.

redcgs.lib

Applies decide to each generator of F, combines the results, and removes repeated generators.

crypto.lib

Convert a hexadecimal number given as a string to its corresponding decimal value (as a bigint).

types.doc / Declaration of objects of a user defined type

Shows how to declare and copy objects of a user-defined type in Singular, including a note about ring requirements when no polynomial field is assigned.

decodegb.lib

Decode a received word using the code's check matrix via a quadratic-equation based method, returning the closest codeword.

decodegb.lib

Decode random received words for a code given by its parity check matrix, for a specified number of trials (optionally fixing correction capacity).

22 commands

This library generates several polynomial systems used for decoding cyclic codes and for computing their minimum distance. In particular, it follows Cooper's philosophy and uses generalized Newton identities. The original method based on quadratic equations is

decodegb.lib

Process random linear codes and attempt decoding of random received words, optionally using a given correction capacity.

decodegb.lib

Generate random linear codes and attempt FL decoding for multiple random received words, printing intermediate Gröbner bases.

crypto.lib

RSA-decodes a ciphertext bigint given as a string and returns the decoded plaintext as a string.

brnoeth.lib

Decode a received word using the basic Skorobogatov–Vladut algorithm with prepSV data, returning a codeword or a 0-matrix if impossible.

math.doc / Decoding codes with Groebner bases

Introduces key notions and results for decoding linear (especially cyclic) codes and computing their minimum distance, with pointers to relevant Singular libraries and references.

math.doc / Decoding method based on quadratic equations

Defines unknown syndromes and an MDS basis, constructs ideals from known syndromes, and states a theorem linking solutions of these ideals to unique decoding up to half the minimum distance.

findifs.lib

Decompose a polynomial into two summands according to whether coefficients involve the given number/parameter.

11 commands

This library implements functional univariate and multivariate decomposition of multivariate polynomials.

ehv.lib

Compute the associated primes of a zero-dimensional radical ideal using the EHV decomposition method.

decomp.lib

Compute a maximal decomposition f=g∘h with g univariate and h a single monomial; returns ideal(g,h) or f if none.

decomp.lib

Tries to decompose a monic multivariate polynomial f as f = g∘h with deg(g)=r; returns ideal(g,h) or f if not decomposable.

decomp.lib

Display current decomp.lib global option settings; optionally reset user-defined options to defaults.

decomp.lib

Decompose a polynomial if it is composite; optionally compute a full decomposition when called with flag 1.

VecField.lib

Decomposes a VecField (or list of VecFields) into semisimple and nilpotent components.

decomp.lib

Try to decompose a univariate monic polynomial f as f=g∘h with deg(g)=r; returns ideal(g,h) or f if impossible.

surfex.lib

Convert a decimal number given as a string (<=30 digits) to a rational-number string with 14-digit precision.

kernel

generic value (any type)

types.doc / def

Describes how untyped objects defined with def inherit their type from the first assignment. Notes special behavior for rings, which are referenced rather than copied.

types.doc / def declarations

Describes the syntax and purpose of def declarations in Singular, including notes on usage and an example.

reference.doc / defined

Returns a non-zero value (TRUE) if a user-defined object with the given name exists, and 0 (FALSE) otherwise. The non-zero value indicates the level where the object is defined.

multigrading.lib

Compute a matrix to append to a grading group matrix so that a polynomial becomes homogeneous.

normaliz.lib

Constructs a toric ring from leading monomials of I and exports the defining binomial ideal J into that ring.

types.doc / Definition of a user defined type

Describes the syntax and purpose of newstruct for defining user-defined types in Singular, including available operations and examples.

numerDecom.lib

Constructs a deflation ring for an ideal I of deflation order d; the deflation ideal is available in the returned ring as DI.

deflation.lib

Deflation algorithm (Hauenstein–Mourrain–Szanto) returning a ring whose exported ideal AUGF is the augmented system.

deflation.lib

Performs one deflation iteration by augmenting F with determinants of selected Jacobian submatrices.

deflation.lib

Constructs an augmented (deflated) polynomial system and returns the ring containing the exported ideal AUGF.

deflation.lib

Construct an augmented (deflated) polynomial system in an extended ring by adding lambda, B, and h variables (LVZ method).

4 commands

This library provides several deflation methods for polynomial systems.

sing.lib

Compute a matrix whose columns form a k-basis of infinitesimal deformations of an ideal or polynomial.

12 commands

## Miniversal Deformation of Singularities and Modules

examples.doc / Deformations

Describes Singular libraries and procedures for computing miniversal (semiuniversal) deformations of isolated singularities, and provides example computations.

tutor.tex / Deformations

Describes Singular libraries and procedures for computing miniversal (semiuniversal) deformations of isolated singularities, and provides example computations in deform.lib and sing.lib.

ring.lib

Create and return a ring with given characteristic, ordering, and n variable names derived from a base string.

ring.lib

Create and return a polynomial ring with characteristic p (default 32003), ordering dp, and n variables.

ring.lib

Create a ring with characteristic p (default 32003), ds ordering, and n variables (x,y,z,... or x(1..n)).

modules.lib

Compute degree like deg(), using the ring's custom grading set by SetDeg if present.

reference.doc / deg

Returns the maximal (weighted) degree of the terms of a polynomial or a vector. By definition, deg(0) is -1.

reference.doc / degBound

Defines a degree limit that can stop certain standard basis computations. The interpretation of “degree” depends on the ring ordering and the command used.

modules.lib

Return a list of the degrees (grading) of a graded free module's generators.

reference.doc / degree

Describes degree, which prints the (Krull) dimension and multiplicity computed from leading monomials of an ideal or module. Includes syntax, type, purpose, and an example.

divisors.lib

Compute the degree of a divisor as mult(std(A.num)) - mult(std(A.den)).

divisors.lib

Compute the degree of a formal divisor as the weighted sum of the degrees of its summands.

presolve.lib

Split generators of an ideal/module into those with (weighted) total degree in [d1,d2] and the remaining ones.

numerAlg.lib

Compute the degree of the pure i-dimensional component of the variety defined by an ideal.

modules.lib

Return the grading (degrees) list of a graded module.

reference.doc / delete

Describes the delete command, which removes elements at specified indices from lists, intvecs, ideals, or modules without changing the input.

redcgs.lib

Deletes sibling vertices under the given vertex when their associated ideals are detected as containing another child's ideal.

dmodapp.lib

Delete the k-th generator from an ideal or module and return the modified object.

general.lib

Delete elements at the positions given by an intvec from a list and return the resulting list.

modstd.lib

Remove unlucky primes/results from paired lists (and optional aligned list) and return filtered lists plus the prevalent leading ideal.

redcgs.lib

Delete a vertex (given by its label intvec) from the global structure @L, recursively deleting its children.

redcgs.lib

Return ideal F with the i-th generator removed (1-based); i=0 returns F unchanged.

redcgs.lib

Return J with all generators that also occur in I removed.

redcgs.lib

Return a copy of an intvec with the i-th entry removed.

hnoether.lib

Compute the delta invariant of an isolated plane curve singularity at 0 from a polynomial or Hamburger-Noether expansion data.

resolve.lib

Compute the Delta-operator applied to the ideal J of a basic object BO (Bravo–Encinas–Villamayor notation).

resolve.lib

Iteratively applies the Delta-operator to the ideal J in a basic object and returns the list of successive Delta^i(J).

normal.lib

Compute sums of local delta invariants, Tjurina numbers, and branch counts of f at singular points described by a prime ideal J.

arnold.lib

Return [d,n] with Delta invariant d and number of branches n for a germ right equivalent to one with nondegenerate Newton boundary.

general.doc / Demo mode

Describes how to run interactive Singular demonstrations using the Emacs interface, including starting, stepping through, exiting, and customization.

reference.doc / denominator

Returns the denominator of a number. The result has type number.

homolog.lib

Compute the depth of coker(M) with respect to the maximal ideal of the basering or a given ideal I.

examples.doc / Depth

Compute the depth of the module of Kähler differentials D_k(R) for a variety defined by (m+1)-minors of a generic symmetric (n\times n) matrix, using a resolution and the Auslander–Buchsbaum formula.

tutor.tex / Depth

Computes the depth of the module of Kähler differentials D_k(R) for a variety defined by (m+1)-minors of a generic symmetric (n\times n) matrix, via a minimal resolution and the Auslander–Buchsbaum formula.

mregular.lib

Compute the depth of S/i for a proper homogeneous ideal i in the current basering (returns -1 on non-homogeneous or i=(1)).

5 commands

A library for computing the de Rham cohomology of complements of complex affine varieties.

dmodapp.lib

Compute a basis of the n-th de Rham cohomology group of the complement of the hypersurface f=0.

dmodapp.lib

Compute an ideal whose entries represent a basis of the n-th de Rham cohomology of the complement of a hypersurface (under assumptions on I).

deRham.lib

Compute the de Rham cohomology groups of the complement of the affine variety defined by polynomials in L.

ainvar.lib

Applies the vector field defined by an nx1 matrix to a poly/vector/ideal componentwise, returning the same type.

difform.lib

Type for derivations, i.e. maps from the degree-1 part of the differential algebra to the basering.

difform.lib

Sets up a differential algebra example ring and exports a sample derivation (phi) and the ring to Top.

difform.lib

Example procedure that sets up a differential algebra ring and exports a derivation phi (and the ring) to Top.

difform.lib

Sets up a diffAlgebra ring and exports three example derivations (phi_1, phi_2, phi_3) to Top.

difform.lib

Compute the sum of two derivations componentwise (assuming matching sorted structure lists).

difform.lib

Validate that a list has the correct structure to define a derivation; throws an error if not.

difform.lib

Constructor for derivations; dispatches to derivationFromList or derivationFromPoly depending on the input type.

difform.lib

Applies the contraction map i_phi of a derivation phi to a differential form df.

difform.lib

Compute the contraction i_phi of a differential algebra generator d_gen with respect to a derivation phi.

difform.lib

Test whether two derivations are equal by comparing generator images componentwise (returns 1 if equal, 0 otherwise).

difform.lib

Evaluates a derivation on a degree-1 (or -1) differential form and returns the resulting polynomial.

difform.lib

Construct a derivation from a validated structure list of generators and their images.

difform.lib

Construct a derivation sending every degree-1 generator of the differential algebra to the given polynomial.

difform.lib

Compute the Lie-derivative L_phi of a differential form df (as i_phi∘d + d∘i_phi).

difform.lib

Compute the componentwise product of two derivations.

difform.lib

Return the negation of a given derivation (unary minus).

difform.lib

Test whether two derivations (or a derivation and a polynomial) are not equal; returns 0 if equal, 1 otherwise.

difform.lib

Prints the given derivation using ls-ordering on the generators.

difform.lib

Compute the componentwise difference of derivations (and supports subtracting a polynomial from a derivation).

difform.lib

Convert a derivation to a string showing images of degree-1 generators, optionally sorted by a given generator monomial ordering.

KVequiv.lib

Compute a matrix whose columns generate derlog(V(iV)), the module of vector fields tangent to the variety defined by iV.

GND.lib

Compute a General Néron Desingularization (per arXiv:1508.05511) from data in the ambient ring, with optional debug/injective flags.

reference.doc / det

Describes the det command, which returns the determinant of a square matrix. The algorithm depends on the input type or an optional second argument.

linalg.lib

Compute the determinant of a matrix using the busadj algorithm.

mondromy.lib

Compute determinant and (if nonzero) adjoint matrix of a square matrix.

realclassify.lib

Returns an upper bound for the determinacy of a polynomial, optionally using a provided Milnor number.

arnold.lib

Determine an upper bound for the determinacy of a polynomial/germ (or return the stored determinacy for a NormalForm).

interval.lib

Compute the determinant of an interval matrix using standard interval arithmetic.

arnold.lib

Determine the Arnold classification of N.phi.sourcegerm (for modality<=2) and store it in N.ArnoldType.

resbinomial.lib

Computes the next blowing-up center and returns lists describing the center and related resolution invariants.

arnold.lib

Compute the exceptional hypersurface of N.normalForm and store it in N.exceptionalHypersurface.

arnold.lib

Update a NormalForm by determining a stable equivalent germ with (semi-)normalized nondegenerate Newton boundary, if possible.

arnold.lib

Compute a normal form and invariants for a polynomial germ with nondegenerate Newton boundary, returning a NormalForm record.

arnold.lib

Compute and store a normal form equation (and parameter-defining extensions) for the polynomial represented by a NormalForm.

tropical.lib

Compute the detropicalisation of a (list of) tropical-style linear polynomial(s), ignoring constant parts.

hnoether.lib

Compute the Hamburger–Noether expansion of an (as power series) irreducible bivariate polynomial, optionally controlling the last line length.

Development-model.md

Describes the git-based development workflow for the Singular Sources repository, including the use of personal forks and how pull requests are handled.

math.doc / Di Biase and Urbanke

Description of the Di Biase and Urbanke algorithm for computing a generating set of a toric ideal, including a key saturation observation and the successive “variable flip” procedure.

matrix.lib

Construct a diagonal matrix either as p times the n×n unit matrix or from all entries of a matrix in row order.

linalg.lib

Test whether a constant square matrix is diagonalizable over the ground field (returns 1/0 or -1 if undecidable/invalid input).

normaliz.lib

Compute the invariant ring of a diagonally acting torus times finite abelian group, returning an ideal of monomial generators.

interval.lib

Create an n×n interval matrix with all diagonal entries set to the given interval.

VecField.lib

Compute a transformation matrix U that simultaneously diagonalizes all matrices in a list.

VecField.lib

Simultaneously diagonalize a list of VecFields (assumes they are simultaneously diagonalizable).

VecField.lib

Change coordinates so the linear parts of a list of VecFields become simultaneously diagonal in the same coordinates.

grobcov.lib

Compute a list of locally closed sets representing the set difference A B from canonical constructible-set forms.

reference.doc / diff

Describes diff, which computes partial derivatives of polynomial objects, applies ideal-based differential operators, and differentiates rational functions over transcendental extensions.

difform.lib

Provides the differential algebra structure and the differential forms dx_1,...,dx_n.

difform.lib

Example procedure that sets up a ring, initializes diffAlgebra(), and exports objects to Top.

difform.lib

Example procedure that creates a ring, initializes diffAlgebra(), and exports the result to Top.

difform.lib

Example procedure that sets up a ring, initializes diffAlgebra, and exports the result to Top.

difform.lib

Example procedure that constructs a quotient ring and initializes the differential algebra setup, exporting data to Top.

difform.lib

Return the differential algebra (as a new ring) with generator and/or ring-variable monomial orderings changed as specified.

difform.lib

Constructs and exports the differential form generators dx_1,...,dx_n in the current differential algebra ring.

difform.lib

Return generators of the differential algebra (or of a specified graded degree) as a list, sorted by the differential algebra monomial ordering.

difform.lib

Return the structure data of the differential algebra constructed from the current basering.

difform.lib

Compute the image of an ideal under the universal derivation in the current differential algebra basering.

tateProdCplxNegGrad.lib

Differentiate ideal J with respect to ideal I in an anticommutative setting, correcting signs, and return the derivative matrix.

redcgs.lib

Test whether consecutive elements of an ideal differ (returns 1 if no adjacent duplicates, else 0).

difform.lib

Type for differential forms; stores a polynomial in the differential algebra as an attribute.

difform.lib

Example procedure in difform.lib that sets up a ring, initializes the differential algebra, creates a difform, and exports objects to Top.

difform.lib

Internal example procedure in difform.lib that sets up a ring, initializes diffAlgebra, constructs a difform, and exports objects.

54 commands

A library for computations in the differential algebra over a (quotient) ring.

difform.lib

Add two differential forms and return their sum as a differential form.

difform.lib

Decompose a difform into a list of (generator difform, polynomial coefficient) pairs.

difform.lib

Return the degree of a differential form relative to the highest generator; returns -1 if the form is 0.

difform.lib

Applies the differential (universal derivation) to a differential form or polynomial, yielding a difform.

difform.lib

Divide two differential forms and return the quotient as a differential form.

difform.lib

Compare two difform objects for equality; returns 1 if equal, 0 otherwise.

difform.lib

Constructs a degree-0 differential form by lifting a polynomial into the current differential algebra.

difform.lib

Convert a difform to a string for generator printing, replacing differential algebra variable prefix 'D' by 'd' (coefficients unsorted).

difform.lib

Decompose a differential form into a list of its homogeneous degree components.

difform.lib

Compare two difforms and return 1 if df is bigger than dg w.r.t. the current or a specified monomial ordering, else 0.

difform.lib

Test whether a difform is a generator of the differential algebra (returns 1 if yes, else 0).

difform.lib

Test whether a difform is homogeneous; returns 1 if homogeneous (including 0), 0 otherwise.

difform.lib

Test whether a difform is homogeneous of a specified degree p (with 0 treated as degree -1).

difform.lib

Compare two difforms by monomial order (optionally after changing differential algebra/ring variable ordering); returns 1 if df<dg else 0.

difform.lib

Test whether a given difform occurs in a list (returns 1 if found, 0 otherwise).

difform.lib

Sorts lists of differential forms and special lists of lists.

difform.lib

Multiply two differential forms and return their product as a differential form.

difform.lib

Return the negation of a differential form.

difform.lib

Negated comparison for difforms: returns 0 if equal, 1 otherwise.

difform.lib

Prints a differential form using difformToString with ds-ordering on the generators.

difform.lib

Compute the difference of two differential forms (df - dg).

difform.lib

Convert a differential form to a string, optionally sorting terms by a specified ordering on the generators.

difform.lib

Return the differential form given by the universal derivation applied to a polynomial.

dmodloc.lib

Differentiate the rational function v[1]/v[2] with respect to variable j, returning the result as a vector.

redcgs.lib

Computes a labeled refinement list from minimal associated primes of N and of L[i]+M (in ring @P), or returns an empty list.

findifs.lib

Return a LaTeX string presenting a difference scheme (from decoef output) in nodal form, with optional parameter and auxiliary-part control.

reference.doc / dim

Describes dim, which computes the (Krull) dimension of an ideal or module, and related variants including quotient and cohomological dimension.

letterplace.doc / dim BR_LETTERPLACE_BR

Computes the Gelfand–Kirillov dimension of the algebra basering/(input ideal) using the Ufnarovskij graph. The input ideal must be given as a two-sided Gröbner basis.

plural.doc / dim BR_PLURAL_BR

Describes dim in Plural, which computes the Gelfand–Kirillov dimension for ideals or modules (and submodules) from leading monomials or a left Gröbner basis.

sing.lib

Return the dimension of the singular locus of an ideal (or polynomial).

gfan.lib

Return the dimension (as int) of a cone, fan, or polytope.

scheme.lib

Compute the dimension of the scheme X.

graal.lib

Return the Krull dimension of the localization A_L of A at the given ideal/bearer L.

sheafcoh.lib

Return the vector space dimension of the graded part of degree d of coker(phi).

sheafcoh.lib

Computes the vector space dimension of the degree d graded part of coker(M) using degree vector w.

sheafcoh.lib

Compute the vector space dimension of H^i(F(d)) on P^n for the coherent sheaf F associated to coker(M).

sheafcoh.lib

Computes the dimension h^i(F(d)) for the sheaf F associated to coker(M), using degree vector w.

monomialideal.lib

Compute the Krull dimension of the affine variety defined by a monomial ideal (returns -1 if not monomial).

schubert.lib

Compute the dimension of a moduli space (stack) of stable maps.

polybori.lib

convert polynomial direct

polybori.lib

Convert a PolyBoRi boolean polynomial to a Singular polynomial.

tateProdCplxNegGrad.lib

Computes the direct sum of two bundles.

spcurve.lib

Compute an ideal describing the discriminant of a versal deformation (codimension 2) by Jacobian minors and elimination.

reszeta.lib

Compute the discrepancy values (a_i) for the exceptional divisors of a given resolution.

grobcov.lib

Compute the factorized discriminant of a quadratic polynomial with respect to a given variable/parameter.

redcgs.lib

Process ideal B under a specification r=[N,W,LN], splitting conditions and returning updated specs, basis and condition polynomial.

redcgs.lib

Processes S-polynomial pairs for an ideal, updating state (N,W,LN) and returning a combined result list.

polybori.lib

Return a string visualization of a ZDD, formatting else-branches on new lines to reflect the ZDD structure.

sheafcoh.lib

Print a diagram of sheaf cohomology h^i(F(t)) and chi(F(t)) from an intmat returned by sheafCoh or sheafCohBGG.

arcpoint.lib

Procedure from arcpoint.lib; no help text available to infer arguments or return type.

hnoether.lib

Display the Hamburger–Noether expansion (HNE) from Hamburger–Noether development data; optionally return a ring containing the ideal HNE.

hnoether.lib

Pretty-prints branch invariants from a plane curve polynomial or from Hamburger–Noether expansion data.

hnoether.lib

Display the multiplicity sequence(s) of a plane curve from a bivariate polynomial or Hamburger–Noether expansion data.

pfd.lib

Print a partial fraction decomposition list (as returned by pfd) in a readable form.

pfd.lib

Print a partial fraction decomposition like displaypfd, but with denominators written out (not indexed).

tropical.lib

Display Puiseux expansion(s) returned by puiseuxExpansion; optionally substitute the expansion into the polynomial ("subst").

tropical.lib

Display the output of tropicalLifting; optionally substitute the lifting into the ideal when called with "subst".

resbinomial.lib

Element-wise division of two equally sized lists of numbers, returning a list of quotients.

jacobson.lib

Divide units out from a diagonal matrix in smith/jacobson output and update the transformation matrices accordingly.

reference.doc / division

Describes the Singular command division, which computes a division with remainder for ideals or modules and returns a standard representation, optionally up to a given weighted degree.

plural.doc / division BR_PLURAL_BR

Describes the Singular command division, which computes a left division with remainder for ideals and modules. Includes syntax variants, returned data, and an example.

hyperel.lib

Computes the divisor of a(x)-b(x)y on the hyperelliptic curve y^2+h(x)y=f(x), returned as a list of points with multiplicities.

divisors.lib

A divisor represented by a tuple of ideals defining its positive and negative parts.

scheme.lib

A divisor on a scheme, given by components (schemes) together with their multiplicities.

divisors.lib

Print a divisor D (as (num) - (den)).

divisors.lib

Add two divisors by multiplying their numerator and denominator ideals.

reszeta.lib

Procedure in reszeta.lib (isolated singularities of surfaces); no further signature details available.

21 commands

We implement a class divisor on an algebraic variety and methods for computations with divisors. Divisors are represented by tuples of ideals defining the positive and the negative part. In particular, we implement the group structure on divisors, computatio

symodstd.lib

Return a list of (largest) primes p with (p-1) divisible by k and not dividing any coefficient of I, optionally extending an existing list.

symodstd.lib

Tests whether (p-1) is divisible by k and whether all relevant numerators/denominators and coefficients of I are nonzero mod p.

hyperel.lib

Returns the remainder R of dividing polynomial f by g (f = ag + R, deg(R) < deg(g)).

stdmodule.lib

Performs one-step reduction of a vector f with respect to a free module G over a subalgebra A (finite SAGBI basis).

dmodapp.lib

Compute a presentation of the localization of D/I w.r.t. f^s and Bernstein polynomial roots with multiplicities.

dmodapp.lib

Compute a Weyl-algebra ring presenting the localization of D/I w.r.t. f^s, returning ring with ideal LD0 and list BS.

dmodloc.lib

Compute an ideal J defining the localization of the D-module D/I at a polynomial f (optionally also return the exponent of the natural map).

plulibs.doc / dmod_lib

Let K be a field of characteristic 0. Given a polynomial ring

33 commands

Let K be a field of characteristic 0. Given a polynomial ring

dmodloc.lib

Computes the natural action of a Weyl-algebra poly/ideal on a polynomial f, returning the same type as the operator.

dmodloc.lib

Compute the natural D-module (Weyl algebra) action of a poly/ideal on the rational function w[1]/w[2].

plulibs.doc / dmodapp_lib

Let K be a field of characteristic 0, R = K[x1,...,xN], and D the Weyl algebra in the variables x1,...,xN,d1,...,dN.

33 commands

Let K be a field of characteristic 0, R = K[x1,...,xN], and D the Weyl algebra in the variables x1,...,xN,d1,...,dN.

dmodloc.lib

Checks that the current basering is a characteristic-0 Weyl algebra (no qring) with variables ordered x(1..n),D(1..n).

plulibs.doc / dmodideal_lib

Let K be a field of characteristic 0. Given a polynomial ring R = K[x_1,...,x_n] and a map given by polynomials F_1,...,F_r in R, one is interested in the R[1/(F_1...F_r)]-module of rank one generated by the symbol F^s = F_1^(s_1)...F_r^(s_r), where s_1,..

7 commands

Let K be a field of characteristic 0. Given a polynomial ring R = K[x_1,...,x_n] and a map given by polynomials F_1,...,F_r in R, one is interested in the R[1/(F_1...F_r)]-module of rank one generated by the symbol F^s = F_1^(s_1)...F_r^(s_r), where s_1,..

plulibs.doc / dmodloc_lib

Let I be a left ideal in the n-th polynomial Weyl algebra D = K[x], and let f be a polynomial in K[x].

25 commands

Let I be a left ideal in the n-th polynomial Weyl algebra D = K[x], and let f be a polynomial in K[x].

nchomolog.lib

Compute a presentation of Ext^i(Ext^i(M,D),D) over the current basering D (default i=floor(nvars(D)/2)).

nchomolog.lib

Return Groebner basis of the difference between M and Ext^n(Ext^n(M,D),D) (dual of dual) in the basering.

plulibs.doc / dmodvar_lib

Let K be a field of characteristic 0. Given a polynomial ring R = K[x_1,...,x_n] and polynomials f_1,...,f_r in R, define F = f_1...f_r and F^s = f_1^(s_1)...f_r^(s_r) for symbolic discrete (i.e. shiftable) variables s_1,...,s_r.

5 commands

Let K be a field of characteristic 0. Given a polynomial ring R = K[x_1,...,x_n] and polynomials f_1,...,f_r in R, define F = f_1...f_r and F^s = f_1^(s_1)...f_r^(s_r) for symbolic discrete (i.e. shiftable) variables s_1,...,s_r.

Generated documentation landing page.

general.doc / Documentation Tool

Describes lib2doc, a utility for generating stand-alone documentation for Singular libraries in various output formats, and explains how to install requirements and generate documentation files.

hyperel.lib

Doubles a semireduced hyperelliptic divisor on the curve y^2+h(x)y=f(x) using a special case of Cantor reduction.

purityfiltration.lib

Compute a presentation matrix for the double Ext module Ext^i(Ext^i(M,D),D) where M = coker(R).

platform.doc / Download instructions

Download and installation locations for Singular source and binary packages, including release sources via HTTP and FTP.

tropicalEllipticCovers.lib

Generate (and optionally open) a LaTeX/TikZ document illustrating all elliptic-curve covers determined by a graph, ordering and degree partition.

tropical.lib

Create and display a TeX/PostScript drawing of the Newton subdivision for a tropical plane curve defined by f.

tropical.lib

Compute and display (or export) a tropical plane curve from a polynomial or tropical data, producing TeX/PS files in /tmp.

tropical.lib

Draw (and optionally display) two tropical plane curves given by polynomials or tropical linear-form lists, with min/max and output options.

resolve.lib

Placeholder for dropDim in resolve.lib; currently not implemented (raises an error).

dmodloc.lib

Compute the singular locus ideal of the D-module D/I in a Weyl algebra basering (char 0, variables x(1..n),D(1..n)).

matrix.lib

Return the direct sum (block-diagonal concatenation) of the given matrices.

brnoeth.lib

Compute a generator matrix of the dual code (equivalently a parity-check matrix) from a numeric generator matrix.

gfan.lib

Compute and return the dual cone of a given cone.

tropical.lib

Compute the defining polynomial of the dual conic of an affine plane conic in variables x and y.

chern.lib

Compute the dual (conjugate) partition of a given integer partition, optionally padded with leading zeros to length N.

schubert.lib

Compute the dual of a partition p with respect to integers k and n.

gfan.lib

Compute and return the dual polytope of a given polytope.

schubert.lib

Return the dual of a sheaf.

dummy.lib

Return the given argument unchanged.

1 commands

Dummy library for testing purposes.

reference.doc / dump

Writes the current Singular session state (all defined variables and their values) to a specified link so it can be restored later with getdump.

fpalgebras.lib

Constructs a letterplace free algebra ring encoding a Dyck group presentation and exports the ideal I of relations.

fpalgebras.lib

Construct a Letterplace ring for a Dyck group presentation and export the defining relations as ideal I.

fpalgebras.lib

Constructs a Letterplace ring for a Dyck group presentation and exports the defining ideal I of relations.

general.doc / Dynamic loading

Describes how Singular can be extended at runtime by loading dynamic modules written in C/C++ or other higher-level languages, and how these modules are handled after loading.

examples.doc / Dynamic modules

Example illustrating how to use dynamic modules in Singular. It shows loading a module and computing partial standard bases while considering different numbers of vector components.

tutor.tex / Dynamic modules

Example illustrating how to use dynamic modules in Singular. It loads a dynamic module and shows how changing the number of considered vector components affects the standard basis computation.

teachstd.lib

Compute the ecart of a polynomial or vector as maxdeg1(f) - maxdeg1(lead(f)).

reference.doc / echo

Documents the local setting echo, which controls whether input is echoed depending on the value of voice. Includes a short example and related references.

resbinomial.lib

Compute the E-coefficient ideal E-Coeff_V(P) for a permissible hypersurface V; returns exponents, coefficients, and a classification.

crypto.lib

Elliptic curve primality proving: returns a primality certificate {L,P,m,q} for N or a message that N is not prime.

resbinomial.lib

Compute the E-order for each term described by Coef/Exp, summing exponents over variables selected by flaglist.

general.doc / Editing input

Describes key bindings for editing and recalling input lines in Singular, and explains history and GNU Readline support on Unix-like systems.

general.doc / Editing SINGULAR input files with Emacs

Guidelines for editing Singular input files and libraries in Emacs using C/C++ mode, including recommended .emacs settings and customization tips.

crypto.lib

Extended gcd for a list: returns coefficients T with sum_i L[i]T[i] = gcd(L[1],...,L[n]) stored as T[n+1].

divisors.lib

Compute an effective divisor linearly equivalent to a given divisor.

zeroset.lib

Compute the polynomial gcd of f and g over Q(a)[x] (outdated; use gcd instead).

normaliz.lib

Compute Ehrhart ring data from an ideal’s leading monomials; returns a list of ideals (or the input ideal if Hilbert basis is blocked).

16 commands

Algorithms for primary decomposition and radical computation, based on ideas of Eisenbud, Huneke, and Vasconcelos.

linalg.lib

Compute eigenvalues of a matrix (assumed to lie in the base field) and return them with their multiplicities.

symodstd.lib

Compute eigenvalues of the generators of an ideal under a given permutation of the basering variables.

JMBTest.lib

Compute an EK couple for two jmp elements; returns a list containing an index and a resulting polynomial/monomial.

JMSConst.lib

Compute EK-related data for a pair of jmp objects, returning a 2-element list with an index and a monomial/poly.

JMBTest.lib

Compute EK polynomials (and their degrees) from EK data and a generator list.

JMSConst.lib

Compute EK polynomials from a list of jmp-polynomial lists and return the EK data together with their degrees.

JMBTest.lib

Compute EK polynomials for all pairs in G and return the EK data together with their degrees.

elementsInTermsOfGeneratorsG25.sing

Singular source defining two permutations as generators and a list of group elements written as words in the generators.

polylib.lib

Return the ideal generated by the elementary symmetric polynomials (in n variables).

polylib.lib

Compute the degree-d elementary symmetric polynomial in var(lindex)..var(hindex) of the current basering.

elim.lib

Eliminate specified variables from an ideal/module using an elimination ring and std/slimgb (optionally computing weights).

10 commands

Library elim.lib provides procedures for elimination, saturation, and blowing up.

elim.lib

Eliminate selected variables from an ideal/module using eliminate, choosing a suitable dp/ls ring ordering automatically.

elim.lib

Eliminate the variables given by an intvec from an ideal/module, without requiring a special monomial ordering.

reference.doc / eliminate

Describes eliminate, which removes specified variables from an ideal or module by intersecting with a subring. Includes syntax variants, notes on Hilbert-driven elimination, and examples.