Finding the key of symmetric cipher takes computing common zero of polynomials, which de ne ideal and corresponding variety, usually considered over algebraically closed eld. The solution is the point of the variety over prime eld; it is de ned by a sum of the polynomial ideal and the eld ideal that de nes prime eld. Some authors use partitioning of this sum and reducing syzygies of polynomial ...

This paper studies the representation of a non-negative polynomial f on a non-compact semi-algebraic set K modulo its KKT (Karush-KuhnTucker) ideal. Under the assumption that f satisfies the boundary Hessian conditions (BHC) at each zero of f in K; we show that f can be represented as a sum of squares (SOS) of real polynomials modulo its KKT ideal if f ≥ 0 on K.

the rings considered in this article are commutative with identity $1neq 0$. by a proper ideal of a ring $r$,  we mean an ideal $i$ of $r$ such that $ineq r$.  we say that a proper ideal $i$ of a ring $r$ is a  maximal non-prime ideal if $i$ is not a prime ideal of $r$ but any proper ideal $a$ of $r$ with $ isubseteq a$ and $ineq a$ is a prime ideal. that is, among all the proper ideals of $r$,...

let m be an artinian module over the commutative ring a (with nonzero identity) and a p spec a be such that a is a finitely generated ideal of a and am = m. also suppose that h = h where h. = m/ (0: a )for i

A complete classification is obtained for all second-order linear recurring sequences uniformly distributed modulo an ideal of a Dedekind domain.