Related papers: On the Polya permanent problem over finite fields

Komj\'ath, Milner, and Polat investigated when a finitary matroid admits a partition into circuits. They defined the class of ``finite matching extendable'' matroids and showed in their compactness theorem that those matroids always admit…

Computing the permanent of a non-negative matrix is a computationally challenging, \#P-complete problem with wide-ranging applications. We introduce a novel permanental analogue of Schur's determinant formula, leveraging a newly defined…

We prove the existence of an effective universal upper bound for the order of any integral periodic orbit of any integral algebraic dynamical system in a fixed ambient space. Using this, we demonstrate the decidability of periodicity in…

We develop several notions of multiplicity for linear factors of multivariable polynomials over different arithmetics (hyperfields). The key example is multiplicities over the hyperfield of signs, which encapsulates the arithmetic of…

Kronecker's Theorem and Rabin's Theorem are fundamental results about computable fields F and the decidability of the set of irreducible polynomials over F. We adapt these theorems to the setting of differential fields K, with constrained…

We consider the problem of determining the maximum number of common zeros in a projective space over a finite field for a system of linearly independent multivariate homogeneous polynomials defined over that field. There is an elaborate…

In this paper, we show that a partitioned formula \phi is dependent if and only if \phi has uniform definability of types over finite partial order indiscernibles. This generalizes our result from a previous paper [1]. We show this by…

Permutation polynomials are an interesting subject of mathematics and have applications in other areas of mathematics and engineering. In this paper, we develop general theorems on permutation polynomials over finite fields. As a…

Suppose $f$ and $g$ are two post-critically finite polynomials of degree $d_1$ and $d_2$ respectively and suppose both of them have a finite super-attracting fixed point of degree $d_0$. We prove that one can always construct a rational map…

In [6] we proved that the universal theory of infinite free lattices is (algorithmically) decidable, leaving open the problem of decidability of the full theory of an (infinite) free lattice. We solve this problem by proving that, for every…

In perturbative expansion of field theories on a non-commutative geometry, it is known that planar diagrams dominate when the non-commutativity parameter $\theta$ goes to infinity. We discuss whether the ``planar dominance'' occurs also in…

We characterize those complete commutative positive linear ordered monoids $W$ such that whenever $f$ is a map from a Cauchy complete $W$-metric space to itself, the existence of a fixed point of $f$ is independent of the background model…

We study cofinal systems of finite subsets of $\omega_1$. We show that while such systems can be NIP, they cannot be defined in an NIP structure. We deduce a positive answer to a question of Chernikov and Simon from 2013: in an NIP theory,…

An infinite set is orbit-finite if, up to permutations of the underlying structure of atoms, it has only finitely many elements. We study a generalisation of linear programming where constraints are expressed by an orbit-finite system of…

We show that if f: X --> Y is a finite, separable morphism of smooth curves defined over a finite field F_q, where q is larger than an explicit constant depending only on the degree of f and the genus of X, then f maps X(F_q) surjectively…

We give some classes of power maps with low $c$-differential uniformity over finite fields of odd characteristic, {for $c=-1$}. Moreover, we give a necessary and sufficient condition for a linearized polynomial to be a perfect $c$-nonlinear…

One important question in algebraic complexity is understanding the complexity of polynomial ideals (Grochow, Bulletin of EATCS 131, 2020). Andrews and Forbes (STOC 2022) studied the determinantal ideals $I^{\det}_{n,m,r}$ generated by the…

In this paper we determine all the bijective linear maps on the space of bounded observables which preserve a fixed moment or the variance. Nonlinear versions of the corresponding results are also presented.

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…

For a field K, rational function phi in K(z) of degree at least two, and alpha in P^1(K), we study the polynomials in K[z] whose roots are given by the solutions to phi^n(z) = alpha, where phi^n denotes the nth iterate of phi. When the…