Zeros of block-symmetric polynomials on Banach spaces

We investigate sets of zeros of block-symmetric polynomials on the direct sums of sequence spaces. Block-symmetric polynomials are more general objects than classical symmetric polynomials.An analogues of the Hilbert Nullstellensatz Theorem for block-symmetric polynomials on $\ell_p(\mathbb{C}^n)=\ell_p \oplus \ldots \oplus \ell_p$ and $\ell_1 \oplus \ell_{\infty}$ is proved. Also, we show that if a polynomial $P$ has a block-symmetric zero set then it must be block-symmetric.

On Generalizations of the Hilbert Nullstellensatz for Infinity Dimensions (A Surwai)

The paper contains a proof of Hilbert Nullstellensatz for the polynomials oninfinite-dimensional complex spaces and for a symmetric and a block-symmetric polynomials

ON G-TYPE DOMAINS

In this paper, we introduce and study the notion of G-type domains (a domain R is G-type if its quotient field is countably generated R-algebra). We extend some of the basic properties of G-domains to G-type domains. It's observed that a prime ideal of R[x1, x2,…,xn,…] is G-type if and only if its contractions in R, R[x1, x2,…,xn] for all n ≥ 1 are G-type. Using this concept we give a natural proof of the well-known Hilbert Nullstellensatz in infinite countable-dimensional spaces. Characterizations of Noetherian G-type domains, Noetherian G-type domains with the countable prime avoidance property are given. As a consequence, we observe that in complete Noetherian semi-local rings, G-type ideals and G-ideals are the same. Rings with countable Noetherian dimension which are direct sum of G-type domains are fully determined. Finally, we characterize Noetherian rings in which G-type ideals are maximal.

Expressing Combinatorial Problems by Systems of Polynomial Equations and Hilbert's Nullstellensatz

Systems of polynomial equations over the complex or real numbers can be used to model combinatorial problems. In this way, a combinatorial problem is feasible (e.g., a graph is 3-colourable, Hamiltonian, etc.) if and only if a related system of polynomial equations has a solution.For an infeasible polynomial system, the (complex) Hilbert Nullstellensatz gives a certificate that the associated combinatorial problem is infeasible. Thus, unless P = NP, there must exist an infinite sequence of infeasible instances of each hard combinatorial problem for which the minimum degree of a Hilbert Nullstellensatz certificate of the associated polynomial system grows.In the first part of the paper, we show that the minimum degree of a Nullstellensatz certificate for the non-existence of a stable set of size greater than the stability number of the graph is the stability number of the graph. Moreover, such a certificate contains at least one term per stable set of G. In contrast, for non-3-colourability, we proved that the minimum degree of a Nullstellensatz certificate is at least four. Our efforts so far have only yielded graphs with Nullstellensatz certificates of precisely that degree.In the second part of this paper, for the purpose of computation, we construct new polynomial encodings for the problems of finding in a graph its longest cycle, the largest planar subgraph, the edge-chromatic number, or the largest k-colourable subgraph. We include some applications to graph theory.

Algebraically closed commutative rings

A commutative ring A is said to be algebraically closed if every finite system of polynomial equations and inequations in one or more variables with coefficients in A which has a solution in some (commutative) extension of A already has a solution in A. Abraham Robinson's study of model-theoretic forcing has provided powerful new tools for the study of algebraically closed structures in general, and will be applied here to the study of algebraically closed commutative rings. Familiarity with the model-theoretic notions connected with the study of algebraically closed structures is assumed; for background consult [1], [2], and [3].Our main results are the following:1. The theory of commutative rings with identity has no model companion in the sense of Robinson.2. The Hilbert Nullstellensatz, suitably formulated for the class of algebraically closed commutative rings, holds for finitely generated polynomial ideals but fails for certain infinitely generated polynomial ideals.3. If A is algebraically closed, then A/rad A need not be algebraically closed as a semiprime ring: If A is finitely generic then A/rad A is algebraically closed as a semiprime ring, but if A is infinitely generic then A/rad A is not algebraically closed as a semiprime ring.