Quotient Rings of Polynomial Rings.
We show an explicit relation between the Chow form and the Grothendieck residue; and we clarify the role that the residue can play in the intersection theory besides its role in the division problem.
Soient un corps commutatif et un idéal de l’anneau des polynômes (éventuellement ). Nous prouvons une conjecture de C. Berenstein - A. Yger qui affirme que pour tout polynôme , élément de la clôture intégrale de l’idéal , on a une représentationoù .
We are concerned with solving polynomial equations over rings. More precisely, given a commutative domain A with 1 and a polynomial equation antn + ...+ a0 = 0 with coefficients ai in A, our problem is to find its roots in A.We show that when A = B[x] is a polynomial ring, our problem can be reduced to solving a finite sequence of polynomial equations over B. As an application of this reduction, we obtain a finite algorithm for solving a polynomial equation over A when A is F[x1, ..., xN] or F(x1,...
We consider Stanley-Reisner rings where is the edge ideal associated to some particular classes of hypergraphs. For instance, we consider hypergraphs that are natural generalizations of graphs that are lines and cycles, and for these we compute the Betti numbers. We also generalize some known results about chordal graphs and study a weak form of shellability.
On the ring of polynomials in n variables over a field special isomorphisms ’s of into are defined which preserve the greatest common divisor of two polynomials. The ring is extended to the ring and the ring of generalized polynomials in such a way that the exponents of the variables are non-negative rational numbers and rational numbers, respectively. The isomorphisms ’s are extended to automorphisms ’s of the ring . Using the property that the isomorphisms ’s preserve GCD it...