Related papers: Polynomials over structured grids

A method is described which allows to evaluate efficiently a polynomial in a (possibly trivial) extension of the finite field of its coefficients. Its complexity is shown to be lower than that of standard techniques when the degree of the…

This paper investigates the independence polynomials arising from iterated strong products of cycle graphs, examining their algebraic symmetries and combinatorial structures. Leveraging modular arithmetic and Galois theory, we establish…

For a finite field of odd number of elements we construct families of permutation binomials and permutation trinomials with one fixed-point (namely zero) and remaining elements being permuted as disjoint cycles of same length. Binomials and…

We investigate the structure of ideals generated by binomials (polynomials with at most two terms) and the schemes and varieties associated to them. The class of binomial ideals contains many classical examples from algebraic geometry, and…

We describe a congruence property of solvable polynomials over Q, based on the irreducibility of cyclotomic polynomials over number fields that meet certain conditions.

We extend the Ax-Katz theorem for a single polynomial from finite fields to the rings Z_m with m composite. This extension not only yields the analogous result, but gives significantly higher divisibility bounds. We conjecture what computer…

This article develops a comprehensive theory of multiary graded polyadic algebras, extending the classical concept of group-graded algebras to higher-arity structures. We introduce the notion of grading by multiary groups and investigate…

The multi-variable Schmidt polynomials are defined by $$ S_n^{(r)}(x_0,\ldots,x_n):=\sum_{k=0}^n {n+k \choose 2k}^{r}{2k\choose k} x_k. $$ We prove that, for any positive integers $m$, $n$, $r$, and $\varepsilon=\pm 1$, all the coefficients…

We introduce a novel representation of structured polynomial ideals, which we refer to as chordal networks. The sparsity structure of a polynomial system is often described by a graph that captures the interactions among the variables.…

We analyze the degree-structure induced by large reducibilities under the Axiom of Determinacy. This generalizes the analysis of Borel reducibilities given in references [1], [6] and [5] e.g. to the projective levels.

Graph clustering is a fundamental problem in unsupervised learning, with numerous applications in computer science and in analysing real-world data. In many real-world applications, we find that the clusters have a significant high-level…

The bipartition polynomial of a graph is a generalization of many other graph polynomials, including the domination, Ising, matching, independence, cut, and Euler polynomial. We show in this paper that it is also a powerful tool for proving…

We continue our study on counting irreducible polynomials over a finite field with prescribed coefficients. We set up a general combinatorial framework using generating functions with coefficients from a group algebra which is generated by…

In the first part of this article, we consider ruled surfaces defined over a finite field; we introduce invariants for them, and describe some explicit contructions that illustrate possible behaviour of these invariants. In the second part,…

This paper develops a theory of polynomial maps from commutative semigroups to arbitrary groups and proves that it has desirable formal properties when the target group is locally nilpotent. We apply this theory to solve Waring's Problem…

In this article, we are interested in the enumeration of Fully Packed Loops configurations on a grid with a given noncrossing matching. These quantities also appear as the groundstate components of the Completely Packed Loops model as…

Given a chiral algebra, we study modules over an arbitrary power of a curve. We describe this category in three different ways: in terms of factorization, in terms of certain chiral operations and as modules for a lie algebra in a certain…

For polynomials and rational maps of fixed degree over a finite field, we bound both the average number of connected components of their functional graphs as well as the average number of periodic points of their associated dynamical…

We study the probability distribution of the number of common zeros of a system of $m$ random $n$-variate polynomials over a finite commutative ring $R$. We compute the expected number of common zeros of a system of polynomials over $R$.…

The modular decomposition is a technique that applies but is not restricted to graphs. The notion of module naturally appears in the proofs of many graph theoretical theorems. Computing the modular decomposition tree is an important…