Maximum Number of Common Zeros of Homogeneous Polynomials over Finite Fields

About two decades ago, Tsfasman and Boguslavsky conjectured a formula for the maximum number of common zeros that $r$ linearly independent homogeneous polynomials of degree $d$ in $m+1$ variables with coefficients in a finite field with $q$ elements can have in the corresponding $m$-dimensional projective space. Recently, it has been shown by Datta and Ghorpade that this conjecture is valid if $r$ is at most $m+1$ and can be invalid otherwise. Moreover a new conjecture was proposed for many values of $r$ beyond $m+1$. In this paper, we prove that this new conjecture holds true for several values of $r$. In particular, this settles the new conjecture completely when $d=3$. Our result also includes the positive result of Datta and Ghorpade as a special case. Further, we determine the maximum number of zeros in certain cases not covered by the earlier conjectures and results, namely, the case of $d=q-1$ and of $d=q$. All these results are directly applicable to the determination of the maximum number of points on sections of Veronese varieties by linear subvarieties of a fixed dimension, and also the determination of generalized Hamming weights of projective Reed-Muller codes.Comment: 15 page

Higher weight spectra of Veronese codes

We study q-ary linear codes C obtained from Veronese surfaces over finite fields. We show how one can find the higher weight spectra of these codes, or equivalently, the weight distribution of all extension codes of C over all field extensions of the field with q elements. Our methods will be a study of the Stanley-Reisner rings of a series of matroids associated to each code CComment: 14 page

A characterization of the finite Veronesean by intersection properties

A combinatorial characterization of the Veronese variety of all quadrics in PG(n, q) by means of its intersection properties with respect to subspaces is obtained. The result relies on a similar combinatorial result on the Veronesean of all conics in the plane PG(2, q) by Ferri, Hirschfeld and Thas, and Thas and Van Maldeghem, and a structural characterization of the quadric Veronesean by Thas and Van Maldeghem

Problems on q-Analogs in Coding Theory

The interest in $q$-analogs of codes and designs has been increased in the last few years as a consequence of their new application in error-correction for random network coding. There are many interesting theoretical, algebraic, and combinatorial coding problems concerning these q-analogs which remained unsolved. The first goal of this paper is to make a short summary of the large amount of research which was done in the area mainly in the last few years and to provide most of the relevant references. The second goal of this paper is to present one hundred open questions and problems for future research, whose solution will advance the knowledge in this area. The third goal of this paper is to present and start some directions in solving some of these problems.Comment: arXiv admin note: text overlap with arXiv:0805.3528 by other author

On a conjecture of Ghorpade, Datta and Beelen for the number of points of varities over finite fields

Consider a finite field $\mathbb{F}_q$ and positive integers $d,m,r$ with $1\leq r\leq \binom{m+d}{d}$. Let $S_d(m)$ be the $\mathbb{F}_q$ vector space of all homogeneous polynomials of degree $d$ in $X_0,\dots,X_m$. Let $e_r(d,m)$ be the maximum number of $\mathbb{F}_q$-rational points in the vanishing set of $W$ as $W$ varies through all subspaces of $S_d(m)$ of dimension $r$. Ghorpade, Datta and Beelen had conjectured an exact formula of $e_r(d,m)$ when $q\geq d+1$. We prove that their conjectured formula is true when $q$ is sufficiently large in terms of $m,d,r$. The problem of determining $e_r(d,m)$ is equivalent to the problem of computing the $r^{th}$ generalized hamming weights of projective the Reed Muller code $PRM_q(d,m)$. It is also equivalent to the problem of determining the maximum number of points on sections of Veronese varieties by linear subvarieties of codimension $r$

A Combinatorial Approach to the Number of Solutions of Systems of Homogeneous Polynomial Equations over Finite Fields

We give a complete conjectural formula for the number er(d, m) of maximum possible Fq-rational points on a projective algebraic variety defined by r linearly independent homogeneous polynomial equations of degree d in m + 1 variables with coefficients in the finite field Fq with q elements, when d < q. It is shown that this formula holds in the affirmative for several values of r. In the general case, we give explicit lower and upper bounds for er(d, m) and show that they are sometimes attained. Our approach uses a relatively recent result, called the projective footprint bound, together with results from extremal com-binatorics such as the Clements–Lindström Theorem and its variants. Applications to the problem of determining the generalized Hamming weights of projective Reed–Muller codes are also included. © 2022 Independent University of Moscow

Variety Membership Testing in Algebraic Complexity Theory

In this thesis, we study some of the central problems in algebraic complexity theory through the lens of the variety membership testing problem. In the first part, we investigate whether separations between algebraic complexity classes can be phrased as instances of the variety membership testing problem. For this, we compare some complexity classes with their closures. We show that monotone commutative single-(source, sink) ABPs are closed. Further, we prove that multi-(source, sink) ABPs are not closed in both the monotone commutative and the noncommutative settings. However, the corresponding complexity classes are closed in all these settings. Next, we observe a separation between the complexity class VQP and the closure of VNP. In the second part, we cover the blackbox polynomial identity testing (PIT) problem, and the rank computation problem of symbolic matrices, both phrasable as instances of the variety membership testing problem. For the blackbox PIT, we give a randomized polynomial time algorithm that uses the number of random bits that matches the information-theoretic lower bound, differing from it only in the lower order terms. For the rank computation problem, we give a deterministic polynomial time approximation scheme (PTAS) when the degrees of the entries of the matrices are bounded by a constant. Finally, we show NP-hardness of two problems on 3-tensors, both of which are instances of the variety membership testing problem. The first problem is the orbit closure containment problem for the action of GLk x GLm x GLn on 3-tensors, while the second problem is to decide whether the slice rank of a given 3-tensor is at most r