### D.4.26 pointid_lib

Library:
  pointid.lib
Purpose:
  Procedures for computing a factorized lex GB of the vanishing ideal of a set of points via the Axis-of-Evil Theorem (M.G. Marinari, T. Mora)

Author:
  Stefan Steidel, steidel@mathematik.uni-kl.de

Overview:
  The algorithm of Cerlienco-Mureddu [Marinari M.G., Mora T., A remark on a 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 vanishing ideal of A and N = Mon(x(1),...,x(n)) {LM(f)|f in I(A)} is the 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
  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
  fi = (product of y_mdi) modulo (f1,...,f(i-1))
  where the product runs over all m=1,...,n; and all d in F.

Procedures:

> D.4.26.1 nonMonomials |  | non-monomials of the vanishing ideal id of a set of points
> D.4.26.2 cornerMonomials |  | corner-monomials of the set of non-monomials N
> D.4.26.3 facGBIdeal |  | GB G of id and linear factors of each element of G

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