Related papers: A quantitative general Nullstellensatz for Jacobso…

We present a uniform framework for establishing Nullstellens\"atze for power series rings using quantifier elimination results for valued fields. As an application we obtain Nullstellens\"atze for $p$-adic power series (both formal and…

Let $R$ be a commutative Noetherian ring of dimension $d$. In this paper, we first show that some power of the cohomology annihilator annihilates the $(d+1)$-th Ext modules for all finitely generated modules when either $R$ admits a…

A Nullstellensatz is a theorem providing information on polynomials that vanish on a certain set: David Hilbert's Nullstellensatz (1893) is a cornerstone of algebraic geometry, and Noga Alon's Combinatorial Nullstellensatz (1999) is a…

Let X be a finite set and let k be a commutative ring. We consider the k-algebra of the monoid of all relations on X, modulo the ideal generated by the relations factorizing through a set of cardinality strictly smaller than Card(X), called…

Applying techniques similar to Combinatorial Nullstellensatz we prove a lower estimate of $|f(A,B)|$ for finite subsets $A$, $B$ of a field, and polynomial $f(x,y)$ of the form $f(x,y)=g(x)+yh(x)$, where degree of $g$ is greater then degree…

The famous Jacobian conjecture asks if an endomorphism $f$ of $K[x,y]$ ($K$ is a characteristic zero field) having a non-zero scalar Jacobian is invertible. Let $\alpha$ be the exchange involution on $K[x,y]$: $\alpha(x)= y$ and $\alpha(y)=…

It is proved that the Jacobian of a k-endomorphism of k[x_1,...,x_n] over a field k of characteristic zero taking every tame coordinate to a coordinate, must be a nonzero constant in k. It is also proved that the Jacobian of an…

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.

In this paper we seek to determine the Jacobson radical of certain algebras based on semigroups, and in particular on the semigroups $(\beta S, \Box)$, where $S$ is a cancellative, countable, abelian semigroup and $\beta S$ is its…

We prove a general version of Bezout's form of the Nullstellensatz for arbitrary fields. The corresponding sufficient and necessary condition only involves the local existence of multi-valued roots for each of the polynomials belonging to…

A result of A. Joseph says that any nilpotent or semisimple element $z$ in the Weyl algebra $A_1$ over some algebracally closed field $K$ of characterstic 0 has a normal form up to the action of the automorphism group of $A_1$. It is shown…

We present in this paper a geometric theorem which clarifies and extends in several directions work of Brownawell, Kollar and others on the effective Nullstellensatz. To begin with, we work on an arbitrary smooth complex projective variety…

Let p be a prime and let A=(a_1,...,a_l) be a sequence of nonzero elements in F_p. In this paper, we study the set of all 0-1 solutions to the equation a_1 x_1 + ... + a_l x_l = 0. We prove that whenever l >= p, this set actually…

We establish a relative version of the Nullstellensatz for algebras topologically of finite type over a given Banach Tate ring $A$, under the assumption that the corresponding statement holds for rational localizations of $A$. This applies…

Let $\bar{\Kset}_f$ denote the commutative unital ring of Colombeau's full generalized numbers. This ring can be endowed with an ultra-metric in such a way that it becomes a topological ring. There are many interesting question about…

We prove the following result. Let k be an infinite perfect field of positive characteristic and assume that strong resolution of singularities holds over k. Let R be a localization of a commutative d-dimensional k-algebra of finite type…

We investigate the standard graded $k$-algebras over a field $k$ of characteristic zero for which general linear forms are exact zero divisors. We formulate a conjecture regarding the Hilbert function of such rings. We prove our conjecture…

We introduce and investigate a category-theoretic abstraction of the standard "system-solution" adjunction in affine algebraic geometry. We then look further into these geometric adjunctions at different levels of generality, from syntactic…

Generalised quantum determinantal rings are the analogue in quantum matrices of Schubert varieties. Maximal orders are the noncommutative version of integrally closed rings. In this paper, we show that generalised quantum determinantal…

Let $\alpha,\beta \in \mathbb{R}_{>0}$ be such that $\alpha,\beta$ are quadratic and $\mathbb{Q}(\alpha)\neq \mathbb{Q}(\beta)$. Then every subset of $\mathbb{R}^n$ definable in both $(\mathbb{R},{<},+,\mathbb{Z},x\mapsto \alpha x)$ and…