Let <"@s be a sequential term ordering of the set T of all monomials in the variables x"1,...x"n.The author studies the family A of all ideals in K[x"1,...,x"n], that have the reduced Grobner basis with respect to <"@s with the same associated monomial ideal. It is shown that all such ideals have the same dimension and they are parametrized by an affine scheme V"A over K. Furthermore if <"@s is a degree preserving term ordering on T, then all such ideals have the same Hilbert function. The author also shows that V"A is connected and the set of all prime ideals in A and the set of all smooth ideals in A are in one to one correspondence with open subsets of V"A. Finally it is shown that if J, J'@?A, then Top (J) and Top(J') can have different associated monomial ideals. Since it is possible to find the Top of the monomial ideal associated with J, then it is possible to decide if this ideal is the same as the monomial ideal associated to Top(J).
[1]
William Y. Sit,et al.
Well-ordering of certain numerical polynomials
,
1975
.
[2]
Lorenzo Robbiano,et al.
Term Orderings on the Polynominal Ring
,
1985,
European Conference on Computer Algebra.
[3]
Patrizia M. Gianni,et al.
Gröbner Bases and Primary Decomposition of Polynomial Ideals
,
1988,
J. Symb. Comput..
[4]
D. Bayer.
The division algorithm and the hilbert scheme
,
1982
.
[5]
Bruno Buchberger,et al.
Some properties of Gröbner-bases for polynomial ideals
,
1976,
SIGS.
[6]
Giuseppa Carrà Ferro.
Some properties of the lattice points and their application to differential algebra
,
1987
.