Related papers: A nullstellensatz for linear partial differential …
We give upper bounds for the differential Nullstellensatz in the case of ordinary systems of differential algebraic equations over any field of constants $K$ of characteristic $0$. Let $\vec{x}$ be a set of $n$ differential variables,…
The Weyl closure is a basic operation in algebraic analysis: it converts a system of differential operators with rational coefficients into an equivalent system with polynomial coefficients. In addition to encoding finer information on the…
Hilbert's Nullstellensatz is a fundamental result in algebraic geometry that gives a necessary and sufficient condition for a finite collection of multivariate polynomials to have a common zero in an algebraically closed field. Associated…
Let $K$ be a field of characteristic zero and suppose that $f:\mathbb{N}\to K$ satisfies a recurrence of the form $$f(n)\ =\ \sum_{i=1}^d P_i(n) f(n-i),$$ for $n$ sufficiently large, where $P_1(z),...,P_d(z)$ are polynomials in $K[z]$.…
The (weak) Nullstellensatz over finite fields says that if $P_1,\ldots,P_m$ are $n$-variate degree-$d$ polynomials with no common zero over a finite field $\mathbb{F}$ then there are polynomials $R_1,\ldots,R_m$ such that…
For a finite set $\cal F$ of polynomials over fixed finite prime field of size $p$ containing all polynomials $x^2 - x$ a Nullstellensatz proof of the unsolvability of the system $$ f = 0\ ,\ \mbox{ all } f \in {\cal F} $$ in the field is a…
This paper is a contribution to the study of the geometry of algebras related the Weyl groupoid initiated in \cite{M22}. The Nullstellensatz gives a bijection between radical ideals of such an algebra and their zero loci, the superalgebraic…
Consider the polynomial ring in any finite number of variables over the complex numbers, endowed with the $\ell_1$-norm on the system of coefficients. Its completion is the Banach algebra of power series that converge absolutely on the…
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…
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 show that Hilbert's Nullstellensatz, the problem of deciding if a system of multivariate polynomial equations has a solution in the algebraic closure of the underlying field, lies in the counting hierarchy. More generally, we show that…
In algebraic geometry, one studies the solutions to polynomial equations, or, equivalently, to linear partial differential equations with constant coefficients. These lecture notes address the more general case when the coefficients are…
This article gives a class of Nullstellens\"atze for noncommutative polynomials. The singularity set of a noncommutative polynomial $f=f(x_1,\dots,x_g)$ is $Z(f)=(Z_n(f))_n$, where $Z_n(f)=\{X \in M_n^g: \det f(X) = 0\}.$ The first main…
We prove a Nullstellensatz for the ring of polynomial functions in n non-commuting variables over Hamilton's ring of real quaternions. We also characterize the generalized polynomial identities in n variables which hold over the…
This article extends the classical Real Nullstellensatz to matrices of polynomials in a free $\ast$-algebra $\RR\axs$ with $x=(x_1, \ldots, x_n)$. This result is a generalization of a result of Cimpri\vc, Helton, McCullough, and the author.…
The existence of sufficiently many finite order meromorphic solutions of a differential equation, or difference equation, or differential-difference equation, appears to be a good indicator of integrability. In this paper, we investigate…
Hilbert's Nullstellensatz characterizes polynomials that vanish on the vanishing set of an ideal in C[x]. In the free algebra C<X> the vanishing set of a two-sided ideal I is defined in a dimension-free way using images in…
We give the first known bound for orders of differentiations in differential Nullstellensatz for both partial and ordinary algebraic differential equations. This problem was previously addressed by A. Seidenberg but no complete solution was…
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…
We present a new approach to solving polynomial ordinary differential equations by transforming them to linear functional equations and then solving the linear functional equations. We will focus most of our attention upon the first-order…