Related papers: Gr\"obner bases and final polynomials
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 provide a polynomial time algorithm for computing the universal Gr\"obner basis of any polynomial ideal having a finite set of common zeros in fixed number of variables. One ingredient of our algorithm is an effective construction of the…
Recently Brownawell and the second author proved a "non-degenerate" case of the (unproved) "Zilber Nullstellensatz" in connexion with "Strong Exponential Closure". Here we treat some significant new cases. In particular these settle…
We study almost reverse lexicographic ideals in a polynomial ring over a field of arbitrary characteristic. We give a criterion for a given sequence of nonnegative integers to be the Hilbert function of an almost reverse lexicographic ideal…
We use recent results about linking the number of zeros on algebraic varieties over $\mathbb{C}$, defined by polynomials with integer coefficients, and on their reductions modulo sufficiently large primes to study congruences with products…
We prove the extended Hilbert's Nullstellensatz in the context of Hu-Liu polynomial trirings.
Resultants and Gr\"obner bases are crucial tools in studying polynomial elimination theory. We investigate relations between the variety of the resultant of two polynomials and the variety of the ideal they generate. Then we focus on the…
We discuss the notion of optimal polynomial approximants in multivariable reproducing kernel Hilbert spaces. In particular, we analyze difficulties that arise in the multivariable case which are not present in one variable, for example, a…
The algebra of exponential fields and their extensions is developed. The focus is on ELA-fields, which are algebraically closed with a surjective exponential map. In this context, finitely presented extensions are defined, it is shown that…
In this expositional paper, we discuss commutative algebra -- a study inspired by the properties of integers, rational numbers, and real numbers. In particular, we investigate rings and ideals, and their various properties. After, we…
We introduce a collection of polynomials $F_N$, associated to each positive integer $N$, whose divisibility properties yield a reformulation of the Goldbach conjecture. While this reformulation certainly does not lead to a resolution of the…
We show that the universal Gr\"obner basis and the Graver basis of a binomial edge ideal coincide. We provide a description for this basis set in terms of certain paths in the underlying graph. We conjecture a similar result for a parity…
We generalize signature Gr\"obner bases, previously studied in the free algebra over a field or polynomial rings over a ring, to ideals in the mixed algebra $R[x_1,...,x_k]\langle y_1,\dots,y_n \rangle$ where $R$ is a principal ideal…
Let $k$ be a commutative ring and $S=k[x_0, \ldots, x_n]$ be a polynomial ring over $k$ with a monomial order. For any monomial ideal $J$, there exists an affine $k$-scheme of finite type, called Gr\"obner scheme, which parameterizes all…
Polynomial Systems, or at least their algorithms, have the reputation of being doubly-exponential in the number of variables [Mayr and Mayer, 1982], [Davenport and Heintz, 1988]. Nevertheless, the Bezout bound tells us that that number of…
In this expository paper, we present simple proofs of the Classical, Real, Projective and Combinatorial Nullstellens\"atze. Several applications are also presented such as a classical theorem of Stickelberger for solutions of polynomial…
We study GL-equivariant modules over the infinite variable polynomial ring $S = k[x_1, x_2, ..., x_n, ...]$ with $k$ an infinite field of characteristic $p > 0$. We extend many of Sam--Snowden's far-reaching results from characteristic zero…
This paper studies a class of binomial ideals associated to graphs with finite vertex sets. They generalize the binomial edge ideals, and they arise in the study of conditional independence ideals. A Gr\"obner basis can be computed by…
In this paper we generalize the Ritt-Kolchin method of characteristic sets and the classical Gr\"obner basis technique to prove the existence and obtain methods of computation of multivariate difference-differential dimension polynomials…
Previous work by Mora and Sala provides the reduced Groebner basis of the ideal formed by the elementary symmetric polynomials in $n$ variables of degrees $k=1,\dots,n$, $\langle e_{1,n}(x), \dots, e_{n,n}(x) \rangle$. Haglund, Rhoades, and…