Related papers: A Quaternionic Nullstellensatz

In this paper we prove a strong version of the Hilbert Nullstellensatz in the ring $\mathbb H[q_1,\ldots,q_n]$ of slice regular polynomials in several quaternionic variables. Our proof deeply depends on a detailed analysis of the common…

We study the vanishing sets of slice regular polynomials in several quaternionic variables. We obtain a geometric description of the vanishing sets in two variables, which leads to a new version of the Strong Hilbert Nullstellensatz in the…

Let $K$ be a field and $D$ be a finite-dimensional central division algebra over $K$. We prove a variant of the Nullstellensatz for $2$-sided ideals in the ring of polynomial maps $D^n \to D$. In the case where $D = K$ is commutative, our…

Real Nullstellensatz is a classical result from Real Algebraic Geometry. It has recently been extended to quaternionic polynomials by Alon and Paran. The aim of this paper is to extend their Quaternionic Nullstellensatz to matrix…

Using the Rabinowitsch trick, we prove a version of Nullstellensatz over quaternions, which generalizes Hilbert's Nullstellensatz over complex numbers.

We give a constructive proof of the general Nullstellensatz: a univariate polynomial ring over a commutative Jacobson ring is Jacobson. This theorem implies that every finitely generated algebra over a zero-dimensional ring or the ring of…

Hilbert's Nullstellensatz is one of the most fundamental correspondences between algebra and geometry, and has inspired a plethora of noncommutative analogs. In last two decades, there has been an increased interest in understanding…

We compile a long list of equivalent formulations of Hilbert's Nullstellensatz in infinite dimensions, and prove a persistence result for the strong Nullstellensatz in large polynomial rings.

We present different techniques for applying Combinatorial Nullstellensatz to polynomials over finite fields. For examples, we generalize theorems from Noga Alon's paper on the subject, and present a few of our own.

We survey a few strengthenings and generalizations of the Combinatorial Nullstellensatz of Alon and the Schwartz-Zippel Lemma. These lemmas guarantee the existence of (a certain number of) nonzeros of a multivariate polynomial when the…

Alon's combinatorial Nullstellensatz (Theorem 1.1 from \cite{Alon1}) is one of the most powerful algebraic tools in combinatorics, with a diverse array of applications. Let $\F$ be a field, $S_1,S_2,..., S_n$ be finite nonempty subsets of…

We discuss here some computational aspects of the Combinatorial Nullstellensatz argument. Our main result shows that the order of magnitude of the symmetry group associated with permutations of the variables in algebraic constraints,…

Let R be the ring of polynomials in n central variables over the real quaternion algebra H, and let I be a left ideal in R. We prove that if a polynomial p in R vanishes at all the common zeros of I in H^n with commuting coordinates, then…

In this study we find height bounds for polynomial rings over integral domains. We apply nonstandard methods and hence our constants will be ineffective. Then we find height bounds in the polynomial ring over algebraic numbers to test…

In this paper we prove a Nullstellensatz for supersymmetric polynomials. This gives a bijection between radical ideals and superalgebraic sets. These are algebraic sets which are invariant under the Weyl groupoid of Sergeev and Veselov,…

We present a new effective Nullstellensatz with bounds for the degrees which depend not only on the number of variables and on the degrees of the input polynomials but also on an additional parameter called the {\it geometric degree of the…

We prove different forms of the Nullstellensatz for difference fields and absolutely flat simple difference rings, called pseudofields. A difference ring is a ring on which an arbitrary group is acting by means of ring automorphisms.

We propose polynomial-time algorithms for finding nontrivial zeros of quadratic forms with four variables over rational function fields of characteristic 2. We apply these results to find prescribed quadratic subfields of quaternion…

In this paper, we present the Nullstellensatz in case of the coordinate rings of a nonempty subset of Kn where K is a finite field Fq. Some applications of the Nullstellensatz are also discussed.

Alon's combinatorial Nullstellensatz, and in particular the resulting nonvanishing criterion is one of the most powerful algebraic tools in combinatorics, with many important applications. In this paper we extend the nonvanishing theorem in…