Related papers: A Few More Quadratic APN Functions
Finding functions, particularly permutations, with good differential properties has received a lot of attention due to their varied applications. For instance, in combinatorial design theory, a correspondence of perfect $c$-nonlinear…
Permutation polynomials with coefficients 1 over finite fields attract researchers' interests due to their simple algebraic form. In this paper, we first construct four classes of fractional permutation polynomials over the cyclic subgroup…
We establish bounds on a finite separable extension of function fields in terms of the relative class number, thus reducing the problem of classifying extensions with a fixed relative class number to a finite computation. We also solve the…
Expanding upon recent work, a new class of $A$-functions is introduced that can be viewed as an appropriate generalization of the class of regular $A$-functions, the class of structured $A$-functions, and the class of perfect $A$-functions.…
In this paper we present a new approach to proving some exponential inequalities involving the sinc function. Power series expansions are used to generate new polynomial inequalities that are sufficient to prove the given exponential…
We study the Dehn function at infinity in the mapping class group, finding a polynomial upper bound of degree four. This is the same upper bound that holds for arbitrary right-angled Artin groups.
One fundamental question in the context of the geometric complexity theory approach to the VP vs. VNP conjecture is whether VP = $\overline{\textrm{VP}}$, where VP is the class of families of polynomials that are of polynomial degree and…
We present a new family of sine polynomials that are nonnegative for all $x$ in $[0,\pi]$. We also characterize all nonnegative sine polynomials of degree 3 and all nonnegative cosine polynomials of degree 2. In the latest version, typos in…
We prove a one-parameter family of sharp integral inequalities for functions on the $n$-dimensional unit ball. The inequalities are conformally invariant, and the sharp constants are attained for functions that are equivalent to a constant…
We construct a new infinite family of pairs of imaginary cyclic fields of degree $(p-1)/2$ explicitly with both class numbers divisible by a given prime number $p$. For the proof, we use the fundamental unit of $\mathbb Q(\sqrt{p})$,…
In this note, we construct three new infinite families of surfaces of general type with canonical map of degree 2 onto a surface of general type. For one of these families the canonical system has base points.
We present Euler-type recurrence relations for some partition functions. Some of our results provide new recurrences for the number of unrestricted partitions of $n$, denote by $p(n)$. Others establish recurrences for partition functions…
We study the number of non-isomorphic functional graphs of affine-linear transformations from (\F_q)^n to itself, and we prove upper and lower bounds on this quantity for n large. As a corollary to our result, we prove bounds on the number…
Recent results of Freitas, Kraus, Sengun and Siksek, give sufficient criteria for the asymptotic Fermat's Last Theorem to hold over a specific number field. Those works in turn build on many deep theorems in arithmetic geometry. In this…
Fix a finite collection of primes $\{ p_j \}$, not containing $2$ or $3$. Using some observations which arose from attempts to solve the SIC-POVMs problem in quantum information, we give a simple methodology for constructing an infinite…
In a projective plane PG(2,K) defined over an algebraically closed field K of characteristic 0, we give a complete classification of 3-nets realizing a finite group. An infinite family, due to Yuzvinsky, arises from plane cubics and…
In an earlier paper, we determined the finite fields with indecomposable multiplicative groups and conjectured that there is no infinite field whose multiplicative group is indecomposable. In this paper, we prove this conjecture for several…
Fibonacci numbers can be expressed in terms of multinomial coefficients as sums over integer partitions into odd parts. We use this fact to introduce a family of double inequalities involving the generating function for the number of…
We discuss several enumerative results for irreducible polynomials of a given degree and pairs of relatively prime polynomials of given degrees in several variables over finite fields. Two notions of degree, the {\em total degree} and the…
Permutation rational functions over finite fields have attracted high interest in recent years. However, only a few of them have been exhibited. This article studies a class of permutation rational functions constructed using trace maps on…