Changeset 7de8e4 in git for Singular/LIB/pointid.lib
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 Apr 6, 2009, 11:48:23 AM (14 years ago)
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 (u'spielwiese', '0d6b7fcd9813a1ca1ed4220cfa2b104b97a0a003')
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 1a3911abf7c4f6cd8f297ffcc0771fef2ae9389b
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 0bc582ce3f42d033e854938a1175f56baba17b23
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Singular/LIB/pointid.lib
r0bc582c r7de8e4 11 11 remark by Macaulay or Enhancing Lazard Structural Theorem. Bull. of the 12 12 Iranian Math. Soc., 29 (2003), 103145] associates to each ordered set of 13 points A:={a1,...,as} in K^n, ai:=(ai1,...,ain) 14 @*  a set of monomials N and 15 @* a bijection phi: A > N.16 Here I(A):={f in K[x(1),...,x(n)] f(ai)=0, for all 1<=i<=s} denotes the13 points A:={a1,...,as} in K^n, ai:=(ai1,...,ain)@* 14  a set of monomials N and@* 15  a bijection phi: A > N. 16 Here I(A):={f in K[x(1),...,x(n)]  f(ai)=0, for all 1<=i<=s} denotes the 17 17 vanishing ideal of A and N = Mon(x(1),...,x(n)) \ {LM(f)f in I(A)} is the 18 set of monomials not in the leading ideal of I(A) (w.r.t. the lexicograph. 19 ordering with x(n)>...>x(1)). N is also called the set of nonmonomials of 20 I(A). NOTE: #A = #N and N is a monomial basis of K[x(1..n)]/I(A). 21 In particular, this allows to deduce the set of cornermonomials, i.e. the 22 minimal basis M:={m1,...,mr}, m1<...<mr, of its associated monomial ideal 23 M(I(A)), such that M(I(A))= {k*mik in Mon(x(1),...,x(n)), mi in M} and 24 (by interpolation) the unique reduced lexicographical Groebner basis 25 G := {f1,...,fr} such that LM(fi)=mi for each i, i.e. I(A)=<G>. 18 set of monomials which do not lie in the leading ideal of I(A) (w.r.t. the 19 lexicographical ordering with x(n)>...>x(1)). N is also called the set of 20 nonmonomials of I(A). NOTE: #A = #N and N is a monomial basis of 21 K[x(1..n)]/I(A). In particular, this allows to deduce the set of 22 cornermonomials, i.e. the minimal basis M:={m1,...,mr}, m1<...<mr, of its 23 associated monomial ideal M(I(A)), such that@* 24 M(I(A))= {k*mi  k in Mon(x(1),...,x(n)), mi in M},@* 25 and (by interpolation) the unique reduced lexicographical Groebner basis 26 G := {f1,...,fr} such that LM(fi)=mi for each i, that is, I(A)=<G>. 26 27 Moreover, a variation of this algorithm allows to deduce a canonical linear 27 28 factorization of each element of such a Groebner basis in the sense ot the 28 AxisofEvil Theorem by M.G. Marinari and T. Mora. More precisely :29 Acombinatorial algorithm and interpolation allow to deduce polynomials29 AxisofEvil Theorem by M.G. Marinari and T. Mora. More precisely, a 30 combinatorial algorithm and interpolation allow to deduce polynomials 30 31 @* 31 32 @* y_mdi = x(m)  g_mdi(x(1),...,x(m1)), 32 33 @* 33 i=1,...,r , m=1,...,n,d in a finite indexset F, satisfying34 i=1,...,r; m=1,...,n; d in a finite indexset F, satisfying 34 35 @* 35 36 @* fi = (product of y_mdi) modulo (f1,...,f(i1)) 36 37 @* 37 where the product runs over all m=1,...,n andd in F.38 where the product runs over all m=1,...,n; and all d in F. 38 39 39 40 PROCEDURES: … … 78 79 proc nonMonomials(id) 79 80 "USAGE: nonMonomials(id); id = <list of vectors> or <list of lists> or <module> 80 or <matrix>. 81 or <matrix>.@* 81 82 Let A= {a1,...,as} be a set of points in K^n, ai:=(ai1,...,ain), then 82 A can be given as 83 @*  a list of vectors (the ai are vectors) or 84 @*  a list of lists (the ai are lists of numbers) or 85 @*  a module s.t. the ai are generators or 86 @* a matrix s.t. the ai are columns87 ASSUME: basering must have ordering rp, i.e. of the form 0,x(1..n),rp;83 A can be given as@* 84  a list of vectors (the ai are vectors) or@* 85  a list of lists (the ai are lists of numbers) or@* 86  a module s.t. the ai are generators or@* 87  a matrix s.t. the ai are columns 88 ASSUME: basering must have ordering rp, i.e., be of the form 0,x(1..n),rp; 88 89 (the first entry of a point belongs to the lexsmallest variable, etc.) 89 90 RETURN: ideal, the nonmonomials of the vanishing ideal I(A) of A … … 260 261 The cornermonomials are the leading monomials of an ideal I s.t. N is 261 262 a basis of basering/I. 262 NOTE: In our applications I is the vanishing ideal of a finte set of points.263 NOTE: In our applications, I is the vanishing ideal of a finte set of points. 263 264 EXAMPLE: example cornerMonomials; shows an example 264 265 " … … 273 274 ideal M; 274 275 275 // Test: 1 in N ?, if no, return <1> 276 // Test: Is 1 in N ?, if no, return <1>  276 277 for(i = 1; i <= size(N); i++) 277 278 { … … 373 374 proc facGBIdeal(id) 374 375 "USAGE: facGBIdeal(id); id = <list of vectors> or <list of lists> or <module> 375 or <matrix>. 376 or <matrix>.@* 376 377 Let A= {a1,...,as} be a set of points in K^n, ai:=(ai1,...,ain), then 377 A can be given as 378 @*  a list of vectors (the ai are vectors) or 379 @*  a list of lists (the ai are lists of numbers) or 380 @*  a module s.t. the ai are generators or 381 @* a matrix s.t. the ai are columns382 ASSUME: basering must have ordering rp, i.e. of the form 0,x(1..n),rp;378 A can be given as@* 379  a list of vectors (the ai are vectors) or@* 380  a list of lists (the ai are lists of numbers) or@* 381  a module s.t. the ai are generators or@* 382  a matrix s.t. the ai are columns 383 ASSUME: basering must have ordering rp, i.e., be of the form 0,x(1..n),rp; 383 384 (the first entry of a point belongs to the lexsmallest variable, etc.) 384 385 RETURN: a list where the first entry contains the Groebner basis G of I(A) … … 463 464 poly f; 464 465 ideal G1; // stores the elements of G, i.e. G1 = G the GB of I(A) 465 ideal Y; // stores the linear factors of GBelements in each slope466 ideal Y; // stores the linear factors of GBelements in each loop 466 467 list G2; // contains the linear factors of each element of G 467 468
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