Documentation

Singular libraries, commands, and runnable examples

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Table of contents for the Introduction section of the Singular manual. Links to Background, usage instructions, and Getting started.

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General concepts manual

Table of contents for the “General concepts” document, listing the main sections and their links.

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Mathematical background manual

This chapter introduces key mathematical notions and definitions used throughout the manual and points to further references for details.

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Examples manual

Table of contents for the Examples document, listing the main topics and links to their sections.

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Tutorial Tutorial

Entry link and logo image for the Singular Tutorial.

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Polynomial data manual

Contents for the Polynomial data documentation, listing the main sections of the document.

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Functions and system variables manual

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Index generated

Alphabetical index of topics, libraries, procedures, and types.

Types

Library-defined types

57 unique names (60 definitions across libraries). Type signatures link here for disambiguation.

Kernel types

62 built-in types provided by the Singular kernel.

Bibliography

References extracted from the documentation (deduplicated).

Papers 94

Journal and conference articles cited in help strings.

Chapters 1

Chapters/sections in books and edited volumes.

Books 6

Books, monographs, and edited volumes cited in help strings.

Theses 10

Theses, dissertations, and diploma theses cited in help strings.

Preprints 6

Preprints, arXiv manuscripts, and technical reports.

Library Index

absfact.lib 2 cmds

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

ainvar.lib 9 cmds

## Invariant Rings of the Additive Group

aksaka.lib 8 cmds

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

alexpoly.lib 14 cmds

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 12 cmds

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.

arc.lib 3 cmds

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.

arcpoint.lib 5 cmds

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 $\mathrm{Tr}(i)$.

arnold.lib 36 cmds

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).

arnoldclassify.lib 9 cmds

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

arr.lib 44 cmds

## arr.lib

assprimeszerodim.lib 2 cmds

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.

autgradalg.lib 6 cmds

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.

bfun.lib 13 cmds

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

bimodules.lib 12 cmds

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.

brillnoether.lib 1 cmds

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

brnoeth.lib 13 cmds

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

central.lib 20 cmds

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

chern.lib 61 cmds

## Symbolic Computations with Chern Classes

cimonom.lib 3 cmds

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.

cisimplicial.lib 6 cmds

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:

classify_aeq.lib 9 cmds

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

classify.lib 22 cmds

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

classify2.lib 2 cmds

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 1 cmds

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

classifyci.lib 2 cmds

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.

classifyMapGerms.lib 8 cmds

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

combinat.lib 3 cmds

Some basic combinatorics utilities.

compregb.lib 4 cmds

Experimental implementation of comprehensive Gröbner systems.

control.lib 17 cmds

Algebraic analysis tools for System and Control Theory.

crypto.lib 66 cmds

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

curveInv.lib 4 cmds

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

curvepar.lib 10 cmds

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.

customstd.lib 2 cmds

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

decodegb.lib 22 cmds

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

decomp.lib 11 cmds

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

deflation.lib 4 cmds

This library provides several deflation methods for polynomial systems.

deform.lib 12 cmds

## Miniversal Deformation of Singularities and Modules

deRham.lib 5 cmds

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

difform.lib 54 cmds

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

divisors.lib 21 cmds

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

dmod.lib 33 cmds

Let `K` be a field of characteristic 0. Given a polynomial ring

dmodapp.lib 33 cmds

Let `K` be a field of characteristic 0, `R = K[x1,...,xN]`, and `D` the Weyl algebra in the variables `x1,...,xN,d1,...,dN`.

dmodideal.lib 7 cmds

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,..

dmodloc.lib 25 cmds

Let `I` be a left ideal in the *n*-th polynomial Weyl algebra `D = K[x]<d>`, and let `f` be a polynomial in `K[x]`.

dmodvar.lib 5 cmds

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.

dummy.lib 1 cmds

Dummy library for testing purposes.

ehv.lib 16 cmds

Algorithms for primary decomposition and radical computation, based on ideas of Eisenbud, Huneke, and Vasconcelos.

elim.lib 10 cmds

Library **elim.lib** provides procedures for elimination, saturation, and blowing up.

ellipticcovers.lib 11 cmds

We implement a formula to compute the number of covers of elliptic curves. The formula is obtained by proving mirror symmetry for arbitrary genus by tropical methods in [BBM].

enumpoints.lib 2 cmds

This library provides procedures to enumerate rational points in the zero set of an ideal.

equising.lib 5 cmds

Computations related to equisingularity strata for families of plane curve singularities.

ffmodstd.lib 8 cmds

A library for computing a Groebner basis of an ideal in a polynomial ring over an algebraic function field $\mathbb{Q}(T) := \mathbb{Q}(t_1,\ldots,t_m)$ using modular methods and sparse multivariate rational interpolation, where the $t_i$ are transcendenta

ffsolve.lib 6 cmds

Library **ffsolve.lib** provides procedures for solving systems of multivariate polynomial equations over finite fields. Several methods are available (including Groebner basis and linearization), and **ffsolve** selects a method heuristically.

findifs.lib 10 cmds

We provide a presentation of difference operators in polynomial, semi-factorized, and nodal form. Running `findifs_example();` demonstrates how to generate finite difference schemes for linear PDEs from given approximations.

finitediff.lib 24 cmds

Procedures to compute finite difference schemes.

finvar.lib 49 cmds

A library for computing polynomial invariants of finite matrix groups and generators of related varieties. The algorithms are based on work by B. Sturmfels, G. Kemper, S. King, W. Decker, and others.

fpadim.lib 8 cmds

Given the free associative algebra $A = K\langle x_1,\ldots,x_n\rangle$ and a (finite or truncated) Gröbner basis $GB$, one is interested in the following data for $A/\langle GB\rangle$:

fpalgebras.lib 29 cmds

Generates various finitely presented algebras, including group algebras of finitely presented groups, in the Letterplace ring.

fpaprops.lib 9 cmds

This library implements algorithms to compute various algorithmic ring-theoretic properties of finitely presented algebras.

freegb.lib 25 cmds

For the theory, see the chapter **Letterplace** in the Singular Manual.

general.lib 22 cmds

This library provides elementary utility procedures for general computations, such as combinatorial functions, simple numeric constants, list and string helpers, and small wrappers for common algebraic operations.

gfan.lib 60 cmds

Interface to gfan and gfanlib for computations in convex geometry.

gitfan.lib 39 cmds

This library provides procedures to compute GIT-fans, torus orbits, and GKZ-fans.

gkdim.lib 2 cmds

Procedures for computing the GelfandKirillov dimension.

gmspoly.lib 2 cmds

A library for computing the Gauss-Manin system of a cohomologically tame polynomial $f$. Schulze's algorithm [Sch05], based on Sabbah's theory [Sab98], is used to compute a good basis of (the Brieskorn lattice of) the Gauss-Manin system and the differential

gmssing.lib 21 cmds

A library for computing invariants related to the GaussManin system of an isolated hypersurface singularity.

GND.lib 3 cmds

Provides a method to compute the General Néron Desingularization in the setting of one-dimensional local domains.

goettsche.lib 11 cmds

This library provides implementations of formulas (including Drezet’s formula) for computing Poincaré polynomials and Betti numbers of several moduli spaces, including Hilbert schemes of points, punctual Quot-schemes, symmetric powers, and moduli spaces of Kro

graal.lib 10 cmds

This library provides a computational treatment of localizations at prime ideals and their associated graded rings, based on work of Mora.

gradedModules.lib 36 cmds

The library provides procedures for constructing and manipulating graded modules, graded matrices, and graded resolutions.

graphics.lib 3 cmds

Procedures for generating Mathematica input text to visualize ideals, including staircase diagrams and plots (using *ImplicitPlot*).

grobcov.lib 34 cmds

In 2010, this library was designed to contain the Montes–Wibmer algorithm for computing the **Canonical Gröbner Cover** of a parametric ideal. The central routine is `grobcov`.

groups.lib 1 cmds

Library **groups.lib** provides basic functionality for finite group theory.

grwalk.lib 7 cmds

### Groebner Walk Conversion Algorithms

hdepth.lib 2 cmds

This library provides procedures to compute **hdepth_1** for a module with respect to the $\mathbb{Z}$-grading.

hess.lib 1 cmds

Let f be an absolutely irreducible polynomial in two variables x,y. Assume that f is monic as a polynomial in y. Let F = Quot(k[x,y]/f) be the function field defined by f. Define O_F = IntCl(k[x],F) and O_(F,inf) = IntCl(k[1/x],F). We represent a divisor D on

hnoether.lib 36 cmds

A library for computing the Hamburger–Noether expansion (an analogue of the Puiseux expansion over fields of arbitrary characteristic) of a reduced plane curve singularity, following:

hodge.lib 4 cmds

A library for computing the Hodge ideals [MP19] of $\mathbb{Q}$-divisors associated to a reduced hypersurface $f\in R$.

homolog.lib 22 cmds

Procedures for basic computations in homological algebra, such as $\mathrm{Hom}$, $\mathrm{Ext}$, $\mathrm{Tor}$, tensor products, and related invariants for modules given by presentations (typically as cokernels of matrices).

hyperel.lib 17 cmds

Library **hyperel.lib** provides basic procedures for computations on hyperelliptic curves of the form

idealclass.lib 1 cmds

This library implements a ring-independent ideal type used in `scheme.lib`. You can use `+`, `*`, and `^` for addition, multiplication, and exponentiation.

inout.lib 9 cmds

This library provides procedures for printing and displaying objects in compact or page-width-oriented formats, as well as small utilities for formatting text output and pausing computations for user input.

integralbasis.lib 17 cmds

Given an irreducible polynomial f in two variables defining a plane curve, this library implements algorithms to compute an integral basis of the integral closure of the affine coordinate ring in the corresponding algebraic function field.

interval.lib 47 cmds

Implements interval arithmetic for polynomials.

intprog.lib 1 cmds

## Integer Programming with Groebner Basis Methods

invar.lib 9 cmds

This library provides procedures to compute invariant rings for representations of $SL(n)$ and torus groups. It also includes Reynolds operators for $SL(n)$ and for torus (and finite) groups.

involut.lib 7 cmds

An involution is an anti-automorphism of a noncommutative K-algebra such that applying it twice yields the identity. An involution is linear over the ground field.

jacobson.lib 5 cmds

We work over a ring R which is a Euclidean principal ideal domain.

JMBTest.lib 18 cmds

The library performs the J-marked basis test, as described in [CR] and [BCLR]. The test is based on the criterion in [BCLR] concerning Eliahou–Kervaire polynomials (EK).

JMSConst.lib 23 cmds

The library performs J-marked computations as described in [BCLR]. As in `JMBTest.lib`, it constructs the V-polynomials and reduces the EK-polynomials with respect to them, collecting the coefficients as output.

kskernel.lib 12 cmds

## kskernel.lib — Procedures for computing the kernel

KVequiv.lib 6 cmds

Let (V,0) be a complete intersection singularity in (C^p,0) and f_0:(C^n,0) --> (C^p,0) an analytic map germ, viewed as a section of V, so that the singularity V_0 = f_0^-1(V) is a pullback.

latex.lib 14 cmds

Provides simple support for typesetting Singular objects in **LaTeX2e**. The procedures create LaTeX-formatted strings for various Singular objects and can write basic LaTeX file headers/footers, run LaTeX, and view or clean auxiliary files.

lejeune.lib 2 cmds

Library **lejeune.lib** provides procedures for computations in arc spaces following constructions described by M. Lejeune.

linalg.lib 28 cmds

### Algorithmic Linear Algebra

locnormal.lib 1 cmds

This library is deprecated and is distributed only for compatibility.

lrcalc.lib 6 cmds

An interface to the documented functions of the Littlewood–Richardson Calculator by Anders Buch.

makedbm.lib 5 cmds

This library provides access to a DBM database of singularities used by the Arnold classifier.

mathml.lib 27 cmds

This library provides procedures to convert Singular objects to MathML code and to view the generated MathML output.

matrix.lib 31 cmds

Provides basic and elementary operations on matrices, including construction of special matrices, direct sums and concatenation, submatrices, Gaussian reduction, and elementary row/column operations. It also contains utilities for converting a matrix to an ide

maxlike.lib 5 cmds

This library provides procedures to compute maximum likelihood estimates in algebraic statistics, based on the likelihood ideal and related computations.

methods.lib 9 cmds

Methods select which function to execute based on the types of the input tuple.

moddiq.lib 2 cmds

A library for computing ideal quotients and saturations in a polynomial ring over the rational numbers using modular methods.

modfinduni.lib 2 cmds

This library provides a parallel version of `finduni` via `modular.lib`, using modular methods.

modnormal.lib 0 cmds

This library is deprecated and is distributed only for compatibility.

modprimdec.lib 1 cmds

A library offering a modular version of the primary decomposition algorithm of Gianni, Trager, and Zacharias. The procedures are implemented for use over the rationals.

modquotient.lib 2 cmds

A library for computing ideal quotients and saturation using modular methods.

modstd.lib 9 cmds

A library for computing Groebner bases (standard bases) of ideals and modules in polynomial rings over the rational numbers, using modular methods.

modular.lib 1 cmds

This library provides an abstraction layer for modular techniques, which are well known to speed up many computations and to be easy to parallelize.

modules.lib 53 cmds

This library supports computations with graded free resolutions using a user-defined grading on monomials. For this, it introduces a new class of modules (type `Module`) used by `Resolution`.

modwalk.lib 4 cmds

A library for converting Gröbner bases of an ideal in a polynomial ring over the rational numbers using modular methods. The procedures are inspired by the following paper:

mondromy.lib 7 cmds

A library to compute the monodromy of an isolated hypersurface singularity.

monomialideal.lib 17 cmds

A library for computing primary and irreducible decompositions of a monomial ideal using several methods.

mprimdec.lib 19 cmds

Algorithms for primary decomposition of modules, based on the algorithms of Gianni, Trager and Zacharias and of Shimoyama and Yokoyama (a generalization of the latter was suggested by Hans-Gert Graebe, Leipzig). The library uses elements of `primdec.lib`.

mregular.lib 7 cmds

A library for computing the Castelnuovo–Mumford regularity of a homogeneous ideal **without** requiring the computation of a minimal graded free resolution.

multigrading.lib 53 cmds

This library allows you to (virtually) add multigradings to Singular, i.e. to grade multivariate polynomial rings by arbitrary finitely generated abelian groups.

ncalg.lib 27 cmds

This library provides constructors for selected universal enveloping algebras (G-algebras/GR-algebras) and some related quantum groups, returning the corresponding Singular rings.

ncall.lib 0 cmds

Loads all noncommutative libraries.

ncdecomp.lib 5 cmds

This library provides algorithms for the central character decomposition of a module, i.e. a decomposition into generalized weight modules with respect to the center.

ncfactor.lib 13 cmds

This library implements methods for factoring polynomials in several types of noncommutative algebras, in particular:

ncfrac.lib 13 cmds

This library provides an object-oriented interface to **olga.lib**.

ncHilb.lib 2 cmds

Computes graded and multi-graded Hilbert series of non-commutative algebras using the Letterplace approach.

nchilbert.lib 6 cmds

This library provides procedures to compute Hilbert series, the Hilbert polynomial, and the (Hilbert) multiplicity for modules over noncommutative G-algebras (Plural).

nchomolog.lib 17 cmds

This library provides tools for homological algebra of finitely presented modules over GR-algebras.

ncloc.lib 3 cmds

This library introduces a new type: `ncloc`.

ncModslimgb.lib 1 cmds

## ncModslimgb.lib

ncpreim.lib 7 cmds

In G-algebras, the elimination of variables is more involved than in the commutative case.

ncrat.lib 33 cmds

This library provides a framework for working with non-commutative rational functions (more precisely, rational expressions) and their linear (formal linear) representations.

nctools.lib 27 cmds

General tools for computations in noncommutative algebras (in particular, G-algebras and related constructions).

nets.lib 25 cmds

Nets are arrays of characters that are printed in a matrix format. They can be concatenated horizontally and vertically. When concatenating horizontally, empty rows are filled with spaces. All Singular objects can be converted to a Net by applying the command

nfmodstd.lib 3 cmds

A library for computing a Groebner (standard) basis of an ideal in a polynomial ring over an algebraic number field $\mathbb{Q}(t)$ using modular methods, where $t$ is algebraic over the field of rational numbers $\mathbb{Q}$.

nfmodsyz.lib 2 cmds

A library for computing the syzygy module of a given submodule $I$ of a free module over a polynomial ring with coefficients in an algebraic number field $\mathbb{Q}(t)$, using modular methods. Here $t$ is an algebraic number.

noether.lib 8 cmds

A library for computing the Noether normalization of an ideal **without** requiring the computation of the dimension of the ideal.

normal.lib 23 cmds

# Normalization of Affine Rings

normaliz.lib 37 cmds

The library `normaliz.lib` provides an interface to use Normaliz 3.10.0 (or newer) within SINGULAR. Data are exchanged via files.

ntsolve.lib 3 cmds

## Real Newton Solving of Polynomial Systems

numerAlg.lib 4 cmds

The library contains procedures to

numerDecom.lib 12 cmds

The library provides procedures to compute

olga.lib 21 cmds

Let $A$ be a G-algebra.

orbitparam.lib 5 cmds

This library implements the ChevalleyRosenlicht theorem as stated in Theorem 3.1.4 of [Corwin, Greenleaf].

parallel.lib 5 cmds

This library provides implementations of several parallel “skeletons” (i.e. patterns describing how parallel tasks depend on and interact with each other). It is based on the library `tasks.lib` and is intended both for ordinary Singular users and for authors

paraplanecurves.lib 12 cmds

Suppose $C=\{f(x,y,z)=0\}$ is a rational plane curve, where $f$ is homogeneous of degree $n$ with coefficients in $\mathbb{Q}$ and absolutely irreducible (these conditions are checked automatically).

perron.lib 1 cmds

This library provides a procedure to compute algebraic relations (algebraic dependences) for a given list of polynomials.

pfd.lib 11 cmds

This library implements an algorithm based on the work of E. K. Leinartas to write rational functions in several variables as a sum of terms with “smaller” numerators and denominators.

phindex.lib 4 cmds

This library provides procedures to compute the PoincaréHopf (PH) index of real analytic vector fields, and to compute the signature of a symmetric matrix or quadratic form using the Lagrange method.

pointid.lib 3 cmds

The CerliencoMureddu algorithm (see Marinari, Mora: *A remark on a remark by Macaulay or Enhancing Lazard Structural Theorem*, Bull. Iranian Math. Soc. 29 (2003), 103145) associates to each ordered set of points $A := \{a_1,\ldots,a_s\} \subset K^n$, \(a_i :

polybori.lib 47 cmds

A library for using @code{PolyBoRi} through the SINGULAR interface. It provides procedures to convert structures (polynomials, rings, ideals, and ZDDs) in both directions. In particular, it is possible to compute Boolean Gröbner bases via @ref{boolean_std}. Po

polyclass.lib 12 cmds

This library provides a ring-independent polynomial type `Poly`. It is used as a return type in `classify2.lib`, `realclassify.lib`, and `arnold.lib`. You can use `+`, `*`, and `==` for addition, multiplication, and comparison. The key `over` stores the base r

polylib.lib 21 cmds

Procedures for manipulating polynomials, ideals, and modules.

polymake.lib 10 cmds

Computations with polytopes and fans.

presolve.lib 17 cmds

## Pre-Solving of Polynomial Equations

primdec.lib 31 cmds

Algorithms for primary decomposition based on the ideas of

primdecint.lib 7 cmds

A library for computing the primary decomposition of an ideal in the polynomial ring over the integers, $\mathbb{Z}[x_1,\ldots,x_n]$.

primitiv.lib 3 cmds

## Computing a Primitive Element

puiseuxexpansions.lib 5 cmds

This library implements the Newton–Puiseux algorithm to compute Puiseux expansions. It provides procedures to work with Puiseux expansions in $K[X][Y]$, where $K$ is the field $\mathbb{Q}$ of rational numbers or an algebraic extension of $\mathbb{Q}$.

purityfiltration.lib 16 cmds

Purity is a notion with several meanings. In this library, purity refers to equidimensionality of a module (that is, a module is pure if every nonzero submodule has the same dimension as the module).

qhmoduli.lib 11 cmds

Moduli spaces of semi-quasihomogeneous singularities.

qmatrix.lib 5 cmds

## Quantum matrices, quantum minors, and symmetric groups

random.lib 12 cmds

This library provides procedures to create random and sparse polynomials, ideals, and (integer or polynomial) matrices.

ratgb.lib 1 cmds

Theory: Let A be an operator algebra with @code{R = K[x1,\u2026,xN]} as a subring. The operators are usually denoted by @code{d1,\u2026,dM}.

realclassify.lib 9 cmds

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

realizationMatroids.lib 3 cmds

In tropical geometry, one may ask the following: given a one-dimensional balanced polyhedral fan $C$ that is set-theoretically contained in the tropicalization $\operatorname{trop}(Y)$ of an algebraic variety $Y$, does there exist a curve \(X\subseteq Y\

realrad.lib 4 cmds

Algorithms for computing the real radical of an ideal over the rational numbers and over transcendental extensions of the rational numbers.

recover.lib 17 cmds

This library contains implementations of several algorithms from the paper listed below.

redcgs.lib 83 cmds

## Reduced Comprehensive Gröbner Systems

reesclos.lib 4 cmds

A library to compute the integral closure of an ideal $I$ in a polynomial ring $R = k[x(1),\ldots,x(n)]$ using the Rees algebra $R[It]$ of $I$.

resbinomial.lib 35 cmds

## Combinatorial algorithm of resolution of singularities

resgraph.lib 9 cmds

## Visualization of Resolution Data

resjung.lib 3 cmds

This library provides procedures for resolving surface singularities using Jung's method. It can transform an ideal into a form where its singularities are of Hirzebruch–Jung type or quasi-ordinary, and it can compute a (not strong) resolution of the surface d

resolve.lib 16 cmds

## Resolution of singularities (Desingularization)

resources.lib 4 cmds

The purpose of this library is to manage the computational resources of a Singular session. The library tasks.lib, and any library built upon tasks.lib, respects these settings; i.e., they will not use more computational resources than provided via resources.l

reszeta.lib 13 cmds

### reszeta.lib

ring.lib 32 cmds

This library provides procedures to define, copy, extend, and inspect rings, and to move objects between rings via maps. It also includes helpers to query ring properties (coefficient domain, commutativity, and monomial ordering) and utilities such as tensor p

ringgb.lib 4 cmds

This library provides functions related to coefficient rings, in particular vanishing polynomials and reduction modulo the ideal of vanishing polynomials.

rinvar.lib 16 cmds

Implementation based on Derksen's algorithm. The library was written in the context of the diploma thesis (advisor: Prof. Gert-Martin Greuel) “Computations of moduli spaces of semiquasihomogenous singularities and an implementation in Singular”.

rootisolation.lib 12 cmds

This library uses the interval arithmetic from `interval.so`. It introduces the new type `ivmat`, a matrix whose entries are intervals, implemented as `newstruct`. Various functions for computations with interval matrices are provided, such as Gaussian elimina

rootsmr.lib 15 cmds

Routines for counting the number of real roots of a multivariate polynomial system.

rootsur.lib 17 cmds

Routines to bound and count the number of real roots of a univariate polynomial, using several methods: Descartes' rule of signs, the Budan–Fourier theorem, Sturm sequences, and Sturm–Habicht sequences.

rstandard.lib 4 cmds

Computes Janet bases and border bases for ideals for any monomial ordering, using the idea of r-standard bases (as defined by V. Gerdt).

rwalk.lib 3 cmds

This library provides Groebner walk conversion algorithms for computing a standard basis of an ideal using different randomized walk variants.

sagbi.lib 9 cmds

SAGBI stands for “subalgebra bases analogous to Gröbner bases for ideals”.

sagbigrob.lib 6 cmds

Compute a Sagbi–Gröbner basis of an ideal in a subalgebra.

sagbiNormaliz.lib 5 cmds

The library `sagbiNormaliz.lib` provides functions for computing Sagbi bases of subalgebras $A$ of polynomial rings over a field. It is based on `normaliz.lib`. The available functions compute Sagbi bases either without additional control, degree by degree,

sagbiNormaliz0.lib 5 cmds

The library `sagbiNormaliz0.lib` provides functions for computing SAGBI bases of subalgebras `A` of polynomial rings over a field. It is based on `normaliz.lib`.

scheme.lib 30 cmds

The library implements several new classes centered around schemes.

schreyer.lib 3 cmds

The library contains several procedures for computing (a part of) a Schreyer resolution (cf. [SFO]) and some helpers for derham.lib (which requires resolutions over the homogenized Weyl algebra).

schubert.lib 41 cmds

This library implements classes (variety, sheaf, stack, graph) and methods for computations in intersection theory.

sets.lib 18 cmds

This library implements the type `Set` and basic methods for working with sets.

sheafcoh.lib 18 cmds

Procedures for computing sheaf cohomology.

signcond.lib 3 cmds

Routines to determine the number of solutions of a multivariate polynomial system that satisfy a given sign configuration.

sing.lib 19 cmds

### Invariants of Singularities

sing4ti2.lib 3 cmds

Communication interface to **4ti2**.

SingularityDBM.lib 5 cmds

Library **SingularityDBM.lib** provides access to a database (DBM) of singularities used by the Arnold-Classifier.

solve.lib 12 cmds

### Complex Solving of Polynomial Systems (solve.lib)

spcurve.lib 9 cmds

### Deformations and invariants of CM codimension 2 singularities

spectrum.lib 1 cmds

## Singularity Spectrum for Nondegenerate Singularities

sresext.lib 4 cmds

Procedures for computing Schreyer resolutions of modules over the exterior algebra, and related constructions.

ssi.lib 9 cmds

This library provides helper functions for managing SSI files in Singular.

standard.lib 22 cmds

This section documents procedures from `standard.lib` that are always loaded at start-up.

stanleyreisner.lib 11 cmds

This library implements graded pieces (in a fixed degree) of the cotangent modules $T^1$ and $T^2$ for a Stanley-Reisner ring. Graded pieces of homomorphisms are represented by lists of integers.

stdmodule.lib 9 cmds

Compute standard bases of submodules of a free module over a polynomial subalgebra with respect to the natural grading.

stratify.lib 7 cmds

Algorithmic stratification for unipotent group actions.

surf_jupyter.lib 1 cmds

This library provides procedures for plotting plane curves and surfaces with Surf in Jupyter.

surf.lib 2 cmds

This library provides procedures for plotting plane curves and surfaces with Surf.

surfacesignature.lib 3 cmds

A library for computing the signature of an irreducible surface singularity.

surfex.lib 28 cmds

Procedures for visualizing and rotating surfaces.

surfex.lib 29 cmds

This library provides an interface to the external program **surfex**.

swalk.lib 2 cmds

A library for computing a SAGBI basis of a subalgebra using the SAGBI walk algorithm.

symodstd.lib 10 cmds

A library for computing the Groebner basis (standard basis) of an ideal in a polynomial ring over the rational numbers that is invariant under certain permutations of the variables, using symmetry and modular methods.

systhreads.lib 12 cmds

This library implements basic functionality for shared objects in a multi-threaded system, such as channels, shared tables and lists, and synchronization variables.

tasks.lib 14 cmds

This library provides a parallel framework based on tasks. It introduces a new Singular type @code{task}; an object of this type represents a command (given as a string) applied to a list of arguments. Tasks can be computed in parallel via the procedures in th

tateProdCplxNegGrad.lib 28 cmds

In this library, Tate resolutions are used to compute sheaf cohomology of coherent sheaves on products of projective spaces. The algorithms work for arbitrary products. Computations are carried out over the multigraded Cox ring and the corresponding exterior a

teachstd.lib 14 cmds

This library provides procedures used in teaching and experimenting with standard basis computations, including auxiliary routines (ecart, tails, S-polynomials, criteria, and pair management) and procedures to compute standard bases.

template.lib 3 cmds

This library provides a simple template for a Singular library, including basic example procedures and an info string.

toric.lib 2 cmds

## Standard Basis of Toric Ideals

transformation.lib 1 cmds

The image of a map defined by a matrix is computed up to a specified degree. This can be useful, in particular in Singularity theory, when working with right equivalence.

triang.lib 4 cmds

Decomposes zero-dimensional ideals into triangular sets using several algorithms.

tropical.lib 45 cmds

### Computations in Tropical Geometry

tropicalEllipticCovers.lib 16 cmds

We implement formulas to compute numbers of covers of (abstract) elliptic curves, with and without tangency conditions, and of the surface ExP1.

tropicalNewton.lib 8 cmds

This library contains algorithms for computing

tst.lib 25 cmds

This library provides procedures for running automatic **tst** tests in Singular. It includes helpers to execute shell commands, write test status information, measure run times, and run a few standard algebraic test cases.

VecField.lib 10 cmds

Implements a class `VecField`, represented by a vector.

weierstr.lib 4 cmds

This library provides procedures related to the Weierstrass theorems, in particular Weierstrass division and Weierstrass preparation, for polynomials with respect to the last variable.

zeroset.lib 21 cmds

Algorithms for computing the zero-set of a zero-dimensional ideal in $\mathbb{Q}(a)[x_1,\ldots,x_n]$, as well as roots and factorization of univariate polynomials over $\mathbb{Q}(a)[t]$, where $a$ is an algebraic number.