arXiv subjects

Sandro Mattarei

Publications and source records attributed to Sandro Mattarei.

At least 19 recordsLinked to original sources

arXiv2025

Congruences for sums involving $\binom{rk}{k}$

We primarily investigate congruences modulo $p$ for finite sums of the form $\sum_k\binom{rk}{k}x^k/k$ over the ranges $0<k<p$ and $0<k<p/r$, where $p$ is a prime larger than the positive integer $r$. Here $x$ is an indeterminate, thus allowing specialization to numerical congruences where $x$ takes certain algebraic numbers as values. We employ two different approaches that have complementary strengths. In particular, we obtain congruences modulo $p^2$ for the sum $\sum_{0<k<p}\binom{rk}{k}x^k$, expressed in terms of finite polylogarithms of certain quantities related to $x$.

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arXiv2023

On some coefficients of the Artin-Hasse series modulo a prime

Let $p$ be an odd prime, and let $\sum_{n=0}^{\infty} a_{n}X^{n}\in\mathbb{F}_p[[X]]$ be the reduction modulo $p$ of the Artin-Hasse exponential. We obtain a polynomial expression for $a_{kp}$ in terms of those $a_{rp}$ with $r<k$, for even $k<p^2-1$. A conjectural analogue covering the case of odd $k<p$ can be stated in various polynomial forms, essentially in terms of the polynomial $γ(X) =\sum_{n=1}^{p-2}(B_{n}/n)X^{p-n}$, where $B_n$ denotes the $n$-th Bernoulli number. We prove that $γ(X)$ satisfies the functional equation $γ(X-1)-γ(X)=£_1(X)+X^{p-1}-w_p-1$ in $\mathbb{F}_p[X]$, where $£_1(X)$ and $w_p$ are the truncated logarithm and the Wilson quotient. This is an analogue modulo $p$ of a functional equation, in $\mathbb{Q}[[X]]$, established by Zagier for the power series $\sum_{n=1}^{\infty}(B_{n}/n)X^n$. Our proof of the functional equation establishes a connection with a result of Nielsen of 1915, of which we provide a fresh proof. Our polynomial framing allows us to derive congruences for certain numerical sums involving divided Bernoulli numbers.

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arXiv2023

The Artin-Hasse series and Laguerre polynomials modulo a prime

For an odd prime $p$, let $\mathrm{E}_{p}(X)=\sum_{n=0}^{\infty} a_{n}X^{n}\in\mathbb{F}_p[[X]]$ denote the reduction modulo $p$ of the Artin-Hasse exponential series. It is known that there exists a series $G(X^p)\in \mathbb{F}_{p}[[X]]$, such that $L_{p-1}^{(-T(X))}(X)=\mathrm{E}_{p}(X)\cdot G(X^p)$, where $T(X)=\sum_{i=1}^{\infty}X^{p^{i}}$ and $L_{p-1}^{(α)}(X)$ denotes the (generalized) Laguerre polynomial of degree $p-1$. We prove that $G(X^p)=\sum_{n=0}^{\infty}(-1)^n a_{np}X^{np}$, and show that it satisfies $G(X^p)\,G(-X^p)\,T(X)=X^p. $

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arXiv2023

Cubic rational expressions over a finite field

We study and partially classify cubic rational expressions $g(x)/h(x)$ over a finite field $\mathbb{F}_q$, up to pre- and post-composition with independent Möbius transformations. In particular, we obtain a full classification when $q$ is even, and prove an upper bound of $4q$ for the number of equivalence classes when $q$ is odd.

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arXiv2022

Congruences for partial sums of the generating series for $\binom{3k}{k}$

We produce congruences modulo a prime $p>3$ for sums $\sum_k\binom{3k}{k}x^k$ over ranges $0\le k<q$ and $0\le k<q/3$, where $q$ is a power of $p$. Here $x$ equals either $c^2/(1-c)^3$, or $4s^2/\bigl(27(s^2-1)\bigr)$, where $c$ and $s$ are indeterminates. In the former case we deal more generally with shifted binomial coefficients $\binom{3k+e}{k}$. Our method derives such congruences directly from closed forms for the corresponding series.

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arXiv2021

Irreducible polynomials from a cubic transformation

Let $R(x)=g(x)/h(x)$ be a rational expression of degree three over the finite field $\mathbb{F}_q$. We count the irreducible polynomials in $\mathbb{F}_q[x]$, of a given degree, which have the form $h(x)^{\mathrm{deg}\, f}\cdot f\bigl(R(x)\bigr)$ for some $f(x)\in\mathbb{F}_q[x]$. As an application, we recover the number of irreducible transformation shift registers of order three, previously computed by Jiang and Yang.

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arXiv2021

The earliest diamond of finite type in Nottingham algebras

We prove several structural results on Nottingham algebras, a class of infinite-dimensional, modular, graded Lie algebras, which includes the graded Lie algebra associated to the Nottingham group with respect to its lower central series. Homogeneous components of a Nottingham algebra have dimension one or two, and in the latter case they are called diamonds. The first diamond occurs in degree $1$, and the second occurs in degree $q$, a power of the characteristic. Each diamond past the second is assigned a type, which either belongs to the underlying field or is $\infty$. Nottingham algebras with a variety of diamond patterns are known. In particular, some have diamonds of both finite and infinite type. We prove that each of those known examples is uniquely determined by a certain finite-dimensional quotient. Finally, we determine how many diamonds of type $\infty$ may precede the earliest diamond of finite type in an arbitrary Nottingham algebra.

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arXiv2021

Diamond distances in Nottingham algebras

Nottingham algebras are a class of just-infinite-dimensional, modular, $\mathbb{N}$-graded Lie algebras, which includes the graded Lie algebra associated to the Nottingham group with respect to its lower central series. Homogeneous components of a Nottingham algebra have dimension one or two, and in the latter case they are called diamonds. The first diamond occurs in degree $1$, and the second occurs in degree $q$, a power of the characteristic. Many examples of Nottingham algebras are known, in which each diamond past the first can be assigned a type, either belonging to the underlying field or equal to $\infty$. A prospective classification of Nottingham algebras requires describing all possible diamond patterns. In this paper we establish some crucial contributions towards that goal. One is showing that all diamonds, past the first, of an arbitrary Nottingham algebra $L$ can be assigned a type, in such a way that the degrees and types of the diamonds completely describe $L$. At the same time we prove that the difference in degrees of any two consecutive diamonds in any Nottingham algebra equals $q-1$. As a side-product of our investigation, we classify the Nottingham algebras where all diamonds have type $\infty$.

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arXiv2021

Constituents of graded Lie algebras of maximal class and chain lengths of thin Lie algebras

Thin Lie algebras are infinite-dimensional graded Lie algebras $L=\bigoplus_{i=1}^{\infty}$, with $\dim(L_1)=2$ and satisfying a covering property: for each $i$, each nonzero $z\in L_i$ satisfies $[zL_1]=L_{i+1}$. It follows that each homogeneous components $L_i$ is either one- or two-dimensional, and in the latter case is called a diamond. Hence $L_1$ is a diamond, and if there are no other diamonds then $L$ is a graded Lie algebra of maximal class. We present simpler proofs of some fundamental facts on graded Lie algebras of maximal class, and on thin Lie algebras, based on a uniform method, with emphasis on a polynomial interpretation. Among else, we determine the possible values for the most fundamental parameter of such algebras, which is the dimension of their largest metabelian quotient.

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arXiv2021

A sandwich in thin Lie algebras

A thin Lie algebras is a Lie algebra $L$, graded over the positive integers, with its first homogeneous component $L_1$ of dimension two and generating $L$, and such that each nonzero ideal of $L$ lies between consecutive terms of its lower central series. All its homogeneous components have dimension one or two, and the two-dimensional components are called diamonds. We prove that if the next diamond past $L_1$ of an infinite-dimensional thin Lie algebra $L$ is $L_k$, with $k>5$, then $[Lyy]=0$ for some nonzero element $y$ of $L_1$.

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arXiv2020

Graded Lie algebras of maximal class of type $p$

The algebras of the title are infinite-dimensional graded Lie algebras $L= \bigoplus_{i=1}^{\infty}L_i$, over a field of positive characteristic $p$, that are generated by an element of degree $1$ and an element of degree $p$, and satisfy $[L_i,L_1]=L_{i+1}$ for $i\ge p$. In case $p=2$ such algebras were classified by Caranti and Vaughan-Lee in 2003. We announce an extension of that classification to arbitrary prime characteristic, and prove several major steps in its proof.

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arXiv2019

Graded Lie algebras of maximal class of type $n$

Let $n>1$ be an integer. The algebras of the title, which we abbreviate as algebras of type $n$, are infinite-dimensional graded Lie algebras $L= \bigoplus_{i=1}^{\infty}L_i$, which are generated by an element of degree $1$ and an element of degree $n$, and satisfy $[L_i,L_1]=L_{i+1}$ for $i\ge n$. Algebras of type $2$ were classified by Caranti and Vaughan-Lee in 2000 over any field of odd characteristic. In this paper we lay the foundations for a classification of algebras of arbitrary type $n$, over fields of sufficiently large characteristic relative to $n$. Our main result describes precisely all possibilities for the first constituent length of an algebra of type $n$, which is a numerical invariant closely related to the dimension of its largest metabelian quotient.

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arXiv2018

Generalized finite polylogarithms

We introduce a generalization $£_{d}^{(α)}(X)$ of the finite polylogarithms $£_{d}^{(0)}(X)=£_d(X)=\sum_{k=1}^{p-1}X^k/k^d$, in characteristic $p$, which depends on a parameter $α$. The special case $£_{1}^{(α)}(X)$ was previously investigated by the authors as the inverse, in an appropriate sense, of a parametrized generalization of the truncated exponential which is instrumental in a {\em grading switching} technique for non-associative algebras. Here we extend such generalization to $£_{d}^{(α)}(X)$ in a natural manner, and study some properties satisfied by those polynomials. In particular, we find how the polynomials $£_{d}^{(α)}(X)$ are related to the powers of $£_{1}^{(α)}(X)$ and derive some consequences.

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arXiv2018

A generalized truncated logarithm

We introduce a generalization $G^{(α)}(X)$ of the truncated logarithm $\mathcal{L}_1(X) = \sum_{k=1}^{p-1}X^k/k$ in characteristic $p$, which depends on a parameter $α$. The main motivation of this study is $G^{(α)}(X)$ being an inverse, in an appropriate sense, of a parametrized generalization of the truncated exponential given by certain Laguerre polynomials. Such Laguerre polynomials play a role in a grading switching technique for non-associative algebras, previously developed by the authors, because they satisfy a weak analogue of the functional equation $\exp(X)\exp(Y)=\exp(X+Y)$ of the exponential series. We also investigate functional equations satisfied by $G^{(α)}(X)$ motivated by known functional equations for $\mathcal{L}_1(X)=-G^{(0)}(X)$.

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arXiv2017

Generalizations of self-reciprocal polynomials

A formula for the number of monic irreducible self-reciprocal polynomials, of a given degree over a finite field, was given by Carlitz in 1967. In 2011 Ahmadi showed that Carlitz's formula extends, essentially without change, to a count of irreducible polynomials arising through an arbitrary quadratic transformation. In the present paper we provide an explanation for this extension, and a simpler proof of Ahmadi's result, by a reduction to the known special case of self-reciprocal polynomials and a minor variation. We also prove further results on polynomials arising through a quadratic transformation, and through some special transformations of higher degree.

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arXiv2017

From generating series to polynomial congruences

Consider an ordinary generating function $\sum_{k=0}^{\infty}c_kx^k$, of an integer sequence of some combinatorial relevance, and assume that it admits a closed form $C(x)$. Various instances are known where the corresponding truncated sum $\sum_{k=0}^{q-1}c_kx^k$, with $q$ a power of a prime $p$, also admits a closed form representation when viewed modulo $p$. Such a representation for the truncated sum modulo $p$ frequently bears a resemblance with the shape of $C(x)$, despite being typically proved through independent arguments. One of the simplest examples is the congruence $\sum_{k=0}^{q-1}\binom{2k}{k}x^k\equiv(1-4x)^{(q-1)/2}\pmod{p}$ being a finite match for the well-known generating function $\sum_{k=0}^\infty\binom{2k}{k}x^k= 1/\sqrt{1-4x}$. We develop a method which allows one to directly infer the closed-form representation of the truncated sum from the closed form of the series for a significant class of series involving central binomial coefficients. In particular, we collect various known such series whose closed-form representation involves polylogarithms ${\rm Li}_d(x)=\sum_{k=1}^{\infty}x^k/k^d$, and after supplementing them with some new ones we obtain closed-forms modulo $p$ for the corresponding truncated sums, in terms of finite polylogarithms $£_d(x)=\sum_{k=1}^{p-1}x^k/k^d$.

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arXiv2013

Inversion and subspaces of a finite field

Let $A$ and $B$ two $F_q$-subspaces of a finite field, of the same size, and let $A^{-1}$ denote the set of inverses of the nonzero elements of $A$. Mattarei proved that $A^{-1}$ can only be contained in $A$ if either $A$ is a subfield, or $A$ is the set of trace zero elements in a quadratic extension of a field. Csajbók refined this to the following quantitative statement: if $A^{-1}\not\subseteq B$, then the bound $|A^{-1}\cap B|\le 2|B|/q-2$ holds. He also gave examples showing that his bound is sharp for $|B|\le q^3$. Our main result is a proof of the stronger bound $|A^{-1}\cap B|\le |B|/q\cdot\bigl(1+O_d(q^{-1/2})\bigr)$, for $|B|=q^d$ with $d>3$. We also classify all examples with $|B|\le q^3$ which attain equality in Csajbók's bound.

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arXiv2013

Grading switching for modular non-associative algebras

We describe a grading switching for arbitrary non-associative algebras of prime characteristic p, aimed at producing a new grading of an algebra from a given one. This is inspired by a fundamental tool in the classification theory of modular Lie algebras known as toral switching, which relies on a delicate adaptation of the exponential of a derivation. We trace the development of grading switching, from an early version based on taking the Artin-Hasse exponential of a nilpotent derivation, to a more general version which uses certain generalized Laguerre polynomials playing the role of generalized exponentials. Both versions depend on the existence of appropriate analogues of the functional equation exp(x).exp(y)=exp(x+y) for the classical exponential.

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