Changeset 7de8e4 in git for Singular/LIB/pointid.lib

Timestamp:

Apr 6, 2009, 11:48:23 AM (14 years ago)

Author:

Frank Seelisch <seelisch@…>

Branches:

(u'spielwiese', '0d6b7fcd9813a1ca1ed4220cfa2b104b97a0a003')

Children:

1a3911abf7c4f6cd8f297ffcc0771fef2ae9389b

Parents:

0bc582ce3f42d033e854938a1175f56baba17b23

Message:

removed some docu errors prior to release 3-1-0


git-svn-id: file:///usr/local/Singular/svn/trunk@11625 2c84dea3-7e68-4137-9b89-c4e89433aadc

File:

• Singular/LIB/pointid.lib (modified) (6 diffs)

Singular/LIB/pointid.lib

@@ -11 +11 @@
     remark by Macaulay or Enhancing Lazard Structural Theorem. Bull. of the
     Iranian Math. Soc., 29 (2003), 103-145] associates to each ordered set of
-    points A:={a1,...,as} in K^n, ai:=(ai1,...,ain)
-@*      - a set of monomials N and
-@*      - a bijection phi: A --> N.
-    Here I(A):={f in K[x(1),...,x(n)]|f(ai)=0, for all 1<=i<=s} denotes the
+    points A:={a1,...,as} in K^n, ai:=(ai1,...,ain)@*
+          - a set of monomials N and@*
+          - a bijection phi: A --> N.
+    Here I(A):={f in K[x(1),...,x(n)] | f(ai)=0, for all 1<=i<=s} denotes the
     vanishing ideal of A and N = Mon(x(1),...,x(n)) \ {LM(f)|f in I(A)} is the
-    set of monomials not in the leading ideal of I(A) (w.r.t. the lexicograph.
-    ordering with x(n)>...>x(1)). N is also called the set of non-monomials of
-    I(A). NOTE: #A = #N and N is a monomial basis of K[x(1..n)]/I(A).
-    In particular, this allows to deduce the set of corner-monomials, i.e. the
-    minimal basis M:={m1,...,mr}, m1<...<mr, of its associated monomial ideal
-    M(I(A)), such that M(I(A))= {k*mi|k in Mon(x(1),...,x(n)), mi in M} and
-    (by interpolation) the unique reduced lexicographical Groebner basis
-    G := {f1,...,fr} such that LM(fi)=mi for each i, i.e. I(A)=<G>.
+    set of monomials which do not lie in the leading ideal of I(A) (w.r.t. the
+    lexicographical ordering with x(n)>...>x(1)). N is also called the set of
+    non-monomials of I(A). NOTE: #A = #N and N is a monomial basis of
+    K[x(1..n)]/I(A). In particular, this allows to deduce the set of
+    corner-monomials, i.e. the minimal basis M:={m1,...,mr}, m1<...<mr, of its
+    associated monomial ideal M(I(A)), such that@*
+    M(I(A))= {k*mi | k in Mon(x(1),...,x(n)), mi in M},@*
+    and (by interpolation) the unique reduced lexicographical Groebner basis
+    G := {f1,...,fr} such that LM(fi)=mi for each i, that is, I(A)=<G>.
     Moreover, a variation of this algorithm allows to deduce a canonical linear
     factorization of each element of such a Groebner basis in the sense ot the
-    Axis-of-Evil Theorem by M.G. Marinari and T. Mora. More precisely:
-    A combinatorial algorithm and interpolation allow to deduce polynomials
+    Axis-of-Evil Theorem by M.G. Marinari and T. Mora. More precisely, a
+    combinatorial algorithm and interpolation allow to deduce polynomials
 @*
 @*                y_mdi = x(m) - g_mdi(x(1),...,x(m-1)),
 @*
-    i=1,...,r, m=1,...,n, d in a finite index-set F, satisfying
+    i=1,...,r; m=1,...,n; d in a finite index-set F, satisfying
 @*
 @*                 fi = (product of y_mdi) modulo (f1,...,f(i-1))
 @*
-    where the product runs over all m=1,...,n and d in F.
+    where the product runs over all m=1,...,n; and all d in F.
 
 PROCEDURES:
@@ -78 +79 @@
 proc nonMonomials(id)
 "USAGE:  nonMonomials(id); id = <list of vectors> or <list of lists> or <module>
-         or <matrix>.
+         or <matrix>.@*
          Let A= {a1,...,as} be a set of points in K^n, ai:=(ai1,...,ain), then
-         A can be given as
-@*         - a list of vectors (the ai are vectors) or
-@*         - a list of lists (the ai are lists of numbers) or
-@*         - a module s.t. the ai are generators or
-@*         - a matrix s.t. the ai are columns
-ASSUME:  basering must have ordering rp, i.e. of the form 0,x(1..n),rp;
+         A can be given as@*
+          - a list of vectors (the ai are vectors) or@*
+          - a list of lists (the ai are lists of numbers) or@*
+          - a module s.t. the ai are generators or@*
+          - a matrix s.t. the ai are columns
+ASSUME:  basering must have ordering rp, i.e., be of the form 0,x(1..n),rp;
          (the first entry of a point belongs to the lex-smallest variable, etc.)
 RETURN:  ideal, the non-monomials of the vanishing ideal I(A) of A
@@ -260 +261 @@
          The corner-monomials are the leading monomials of an ideal I s.t. N is
          a basis of basering/I.
-NOTE:    In our applications I is the vanishing ideal of a finte set of points.
+NOTE:    In our applications, I is the vanishing ideal of a finte set of points.
 EXAMPLE: example cornerMonomials; shows an example
 "
@@ -273 +274 @@
   ideal M;
 
-//-------------------- Test: 1 in N ?, if no, return <1> ----------------------
+//-------------------- Test: Is 1 in N ?, if no, return <1> ----------------------
   for(i = 1; i <= size(N); i++)
   {
@@ -373 +374 @@
 proc facGBIdeal(id)
 "USAGE:  facGBIdeal(id); id = <list of vectors> or <list of lists> or <module>
-         or <matrix>.
+         or <matrix>.@*
          Let A= {a1,...,as} be a set of points in K^n, ai:=(ai1,...,ain), then
-         A can be given as
-@*         - a list of vectors (the ai are vectors) or
-@*         - a list of lists (the ai are lists of numbers) or
-@*         - a module s.t. the ai are generators or
-@*         - a matrix s.t. the ai are columns
-ASSUME:  basering must have ordering rp, i.e. of the form 0,x(1..n),rp;
+         A can be given as@*
+          - a list of vectors (the ai are vectors) or@*
+          - a list of lists (the ai are lists of numbers) or@*
+          - a module s.t. the ai are generators or@*
+          - a matrix s.t. the ai are columns
+ASSUME:  basering must have ordering rp, i.e., be of the form 0,x(1..n),rp;
          (the first entry of a point belongs to the lex-smallest variable, etc.)
 RETURN:  a list where the first entry contains the Groebner basis G of I(A)
@@ -463 +464 @@
   poly f;
   ideal G1;          // stores the elements of G, i.e. G1 = G the GB of I(A)
-  ideal Y;           // stores the linear factors of GB-elements in each slope
+  ideal Y;           // stores the linear factors of GB-elements in each loop
   list G2;           // contains the linear factors of each element of G