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Dmitrii V. Pasechnik

Publications and source records attributed to Dmitrii V. Pasechnik.

At least 19 recordsLinked to original sources
arXiv2026

An efficient sum of squares nonnegativity certificate for quaternary quartic

For any 4-variate quartic form $f\geq 0$ (i.e. $f$ nonnegative, homogeneous polynomial of degree $4$ with real coefficients) there exist quadratic forms $q$ and $q'$ so that $qq'f$ is a sum of squares (s.o.s.) of quartics, by reducing to the case of $f=au^2+2bu+c$ with $a$, $b$, $c$ $3$-variate forms of degrees 2, 3, 4, respectively, and invoking on its discriminant $Δ=ac-b^2$ a theorem by Hilbert (1893) asserting that for any ternary sextic $h\geq 0$ there exists a quadric $q''$ so that $q''h$ is s.o.s. of quartics. Towards deciding whether just one $q$ always suffices to make $qf$ a s.o.s, we give explicit examples of non-s.o.s. $f=au^2+2bu+c\geq 0$ with non-s.o.s. $Δ$. However, in all these examples $af$ are s.o.s. That is, the straightforward s.o.s. decomposition via Hilbert (1893) need not be the best possible. While it remains open whether one $q$ always suffices (and we conjecture that $q=a$ suffices), we describe how the existence of such $q$ is related to particular types of s.o.s. decompositions for $Δ$.

math.AG↗
arXiv2025

A database of constructions of Hadamard matrices

Hadamard matrices of order $n$ are conjectured to exist whenever $n$ is $1$, $2$, or a multiple of $4$; a similar conjecture exists for skew Hadamard matrices. We provide constructions covering orders $\le 1208$ of all known Hadamard and skew Hadamard matrices in the open-source software SageMath. This allowed us to verify the correctness of results given in the literature. Within this range, just one order, $292$, of a skew Hadamard matrix claimed to have a known construction, required a fix. We also produce the up to date tables, for $n \le 2999$ (resp. $n\le 999$ for skew case), of the minimum exponents $m$ such that a (skew) Hadamard matrix of order $2^m n$ is known, improving over 100 entries in the previously published sources. We explain how tables' entries are related to Riesel numbers. As a by-product of the latter, we show that the Paley constructions of (skew-)Hadamard matrices do not work for the order $2^m 509203$, for any $m$.

math.CO↗
arXiv2023

On hyperovals in $Q^+(6,4)$

According to a computer search conducted by the author and described in [7], in $Q^+(6, 4)$ there are two types of hyperovals, having 72 and 96 points, respectively. Here we give geometric descriptions for these examples.

math.CO↗
arXiv2023

Implementing Hadamard Matrices in SageMath

Hadamard matrices are $(-1, +1)$ square matrices with mutually orthogonal rows. The Hadamard conjecture states that Hadamard matrices of order $n$ exist whenever $n$ is $1$, $2$, or a multiple of $4$. However, no construction is known that works for all values of $n$, and for some orders no Hadamard matrix has yet been found. Given the many practical applications of these matrices, it would be useful to have a way to easily check if a construction for a Hadamard matrix of order $n$ exists, and in case to create it. This project aimed to address this, by implementing constructions of Hadamard and skew Hadamard matrices to cover all known orders less than or equal to $1000$ in SageMath, an open-source mathematical software. Furthermore, we implemented some additional mathematical objects, such as complementary difference sets and T-sequences, which were not present in SageMath but are needed to construct Hadamard matrices. This also allows to verify the correctness of the results given in the literature; within the $n\leq 1000$ range, just one order, $292$, of a skew Hadamard matrix claimed to have a known construction, required a fix.

math.CO↗
arXiv2021

Splitting fields of real irreducible representations of finite groups

We show that any irreducible representation $ρ$ of a finite group $G$ of exponent $n$, realisable over $\mathbb{R}$, is realisable over the field $E:=\mathbb{Q}(ζ_n)\cap\mathbb{R}$ of real cyclotomic numbers of order $n$, and describe an algorithmic procedure transforming a realisation of $ρ$ over $\mathbb{Q}(ζ_n)$ to one over $E$.

math.RT↗
arXiv2017

On moments of a polytope

We show that the multivariate generating function of appropriately normalized moments of a measure with homogeneous polynomial density supported on a compact polytope P in R^d is a rational function. Its denominator is the product of linear forms dual to the vertices of P raised to the power equal to the degree of the density function. Using this, we solve the inverse moment problem for the set of, not necessarily convex, polytopes having a given set S of vertices. Under a weak non-degeneracy assumption we also show that the uniform measure supported on any such polytope is a linear combination of uniform measures supported on simplices with vertices in S.

math.MG↗
arXiv2017

Locally toroidal polytopes of rank 6 and sporadic groups

We augment the list of finite universal locally toroidal regular polytopes of type {3,3,4,3,3} due to P.McMullen and E.Schulte, adding as well as removing entries. This disproves a related long-standing conjecture. Our new universal polytope is related to a well-known Y-shaped presentation for the sporadic simple group $Fi_{22}$, and admits $S_4\times O_8^+(2){:}S_3$ as the automorphism group. We also discuss further extensions of its quotients in the context of Y-shaped presentations. As well, we note that two known examples of finite universal polytopes of type {3,3,4,3,3} are related to Y-shaped presentations of orthogonal groups over GF(2). Mixing construction is used in a number of places to describe covers and 2-covers.

math.GR↗
arXiv2016

Random Chain Complexes

We study random, finite-dimensional, ungraded chain complexes over a finite field and show that for a uniformly distributed differential a complex has the smallest possible homology with the highest probability: either zero or one-dimensional homology depending on the parity of the dimension of the complex. We prove that as the order of the field goes to infinity the probability distribution concentrates in the smallest possible dimension of the homology. On the other hand, the limit probability distribution, as the dimension of the complex goes to infinity, is a super-exponentially decreasing, but strictly positive, function of the dimension of the homology.

math.CO↗
arXiv2016

Implementing Brouwer's database of strongly regular graphs

Andries Brouwer maintains a public database of existence results for strongly regular graphs on $n\leq 1300$ vertices. We implemented most of the infinite families of graphs listed there in the open-source software Sagemath, as well as provided constructions of the "sporadic" cases, to obtain a graph for each set of parameters with known examples. Besides providing a convenient way to verify these existence results from the actual graphs, it also extends the database to higher values of $n$.

math.CO↗
arXiv2014

Sandpile groups of generalized de Bruijn and Kautz graphs and circulant matrices over finite fields

A maximal minor $M$ of the Laplacian of an $n$-vertex Eulerian digraph $Γ$ gives rise to a finite group $\mathbb{Z}^{n-1}/\mathbb{Z}^{n-1}M$ known as the sandpile (or critical) group $S(Γ)$ of $Γ$. We determine $S(Γ)$ of the generalized de Bruijn graphs $Γ=\mathrm{DB}(n,d)$ with vertices $0,\dots,n-1$ and arcs $(i,di+k)$ for $0\leq i\leq n-1$ and $0\leq k\leq d-1$, and closely related generalized Kautz graphs, extending and completing earlier results for the classical de Bruijn and Kautz graphs. Moreover, for a prime $p$ and an $n$-cycle permutation matrix $X\in\mathrm{GL}_n(p)$ we show that $S(\mathrm{DB}(n,p))$ is isomorphic to the quotient by $\langle X\rangle$ of the centralizer of $X$ in $\mathrm{PGL}_n(p)$. This offers an explanation for the coincidence of numerical data in sequences A027362 and A003473 of the OEIS, and allows one to speculate upon a possibility to construct normal bases in the finite field $\mathbb{F}_{p^n}$ from spanning trees in $\mathrm{DB}(n,p)$.

math.CO↗
arXiv2014

Computing symmetry groups of polyhedra

Knowing the symmetries of a polyhedron can be very useful for the analysis of its structure as well as for practical polyhedral computations. In this note, we study symmetry groups preserving the linear, projective and combinatorial structure of a polyhedron. In each case we give algorithmic methods to compute the corresponding group and discuss some practical experiences. For practical purposes the linear symmetry group is the most important, as its computation can be directly translated into a graph automorphism problem. We indicate how to compute integral subgroups of the linear symmetry group that are used for instance in integer linear programming.

math.CO↗
arXiv2013

Critical groups of generalized de Bruijn and Kautz graphs and circulant matrices over finite fields: an extended abstract

We determine the critical groups of the generalized de Bruijn graphs DB$(n,d)$ and generalized Kautz graphs Kautz$(n,d)$, thus extending and completing earlier results for the classical de Bruijn and Kautz graphs. Moreover, for a prime $p$ the critical groups of DB$(n,p)$ are shown to be in close correspondence with groups of $n\times n$ circulant matrices over $\mathbb{F}_p$, which explains numerical data in [OEIS:A027362], and suggests the possibility to construct normal bases in $\mathbb{F}_{p^n}$ from spanning trees in DB$(n,p)$.

math.CO↗
arXiv2013

CSS-like Constructions of Asymmetric Quantum Codes

Asymmetric quantum error-correcting codes (AQCs) may offer some advantage over their symmetric counterparts by providing better error-correction for the more frequent error types. The well-known CSS construction of $q$-ary AQCs is extended by removing the $\F_{q}$-linearity requirement as well as the limitation on the type of inner product used. The proposed constructions are called CSS-like constructions and utilize pairs of nested subfield linear codes under one of the Euclidean, trace Euclidean, Hermitian, and trace Hermitian inner products. After establishing some theoretical foundations, best-performing CSS-like AQCs are constructed. Combining some constructions of nested pairs of classical codes and linear programming, many optimal and good pure $q$-ary CSS-like codes for $q \in {2,3,4,5,7,8,9}$ up to reasonable lengths are found. In many instances, removing the $\F_{q}$-linearity and using alternative inner products give us pure AQCs with improved parameters than relying solely on the standard CSS construction.

cs.IT↗
arXiv2012

Book drawings of complete bipartite graphs

A "book" with k pages consists of a straight line (the "spine") and k half-planes (the "pages"), such that the boundary of each page is the spine. If a graph is drawn on a book with k pages in such a way that the vertices lie on the spine, and each edge is contained in a page, the result is a k-page book drawing (or simply a k-page drawing). The pagenumber of a graph G is the minimum k such that G admits a k-page embedding (that is, a k-page drawing with no edge crossings). The k-page crossing number nu_k(G) of G is the minimum number of crossings in a k-page drawing of G. We investigate the pagenumbers and k-page crossing numbers of complete bipartite graphs. We find the exact pagenumbers of several complete bipartite graphs, and use these pagenumbers to find the exact k-page crossing number of K_{k+1,n} for 3<=k<=6. We also prove the general asymptotic estimate lim_{k->oo} lim_{n->oo} nu_k(K_{k+1,n})/(2n^2/k^2)=1. Finally, we give general upper bounds for nu_k(K_{m,n}), and relate these bounds to the k-planar crossing numbers of K_{m,n} and K_n.

math.CO↗
arXiv2012

Improved lower bounds on book crossing numbers of complete graphs

A "book with k pages" consists of a straight line (the "spine") and k half-planes (the "pages"), such that the boundary of each page is the spine. If a graph is drawn on a book with k pages in such a way that the vertices lie on the spine, and each edge is contained in a page, the result is a k-page book drawing (or simply a k-page drawing). The k-page crossing number nu_k(G) of a graph G is the minimum number of crossings in a k-page drawing of G. In this paper we investigate the k-page crossing numbers of complete graphs K_n. We use semidefinite programming techniques to give improved lower bounds on nu_k(K_n) for various values of k. We also use a maximum satisfiability reformulation to calculate the exact value of nu_k(K_n) for several values of k and n. Finally, we investigate the best construction known for drawing K_n in k pages, calculate the resulting number of crossings, and discuss this upper bound in the light of the new results reported in this paper.

math.CO↗
arXiv2011

Improved lower bounds for the 2-page crossing numbers of K_{m,n} and K_n via semidefinite programming

It has been long conjectured that the crossing numbers of the complete bipartite graph K_{m,n} and of the complete graph K_n equal Z(m,n) (the value conjectured by Zarankiewicz, who came up with a drawing reaching this value) and Z(n) :=Z(n,n-2)/4, respectively. In a 2-page drawing of a graph, the vertices are drawn on a straight line (the spine), and each edge is contained in one of the half-planes of the spine. The 2-page crossing number v_2(G) of a graph G is the minimum number of crossings in a 2-page drawing of G. Somewhat surprisingly, there are 2-page drawings of K_{m,n} (respectively, K_n) with exactly Z(m, n) (respectively, Z(n)) crossings, thus yielding the conjectures (I) v_2(Km,n) =Z(m,n), and (II) v_2(Kn) = Z(n). It is known that (I) holds for min{m, n} <=6, and that (II) holds for n<=14. In this paper we prove that (I) holds asymptotically (that is, lim_n v_2 (K_{m,n})/Z (m, n) = 1) for m=7 and 8. We also prove (II) for 15<=n<=18 and n=20,24, and establish the asymptotic estimate lim_n v_2(K_n)/Z(n) >= 0.9253. The previous best-known lower bound involved the constant 0.8594.

math.CO↗
arXiv2011

Two distance-regular graphs

We construct two families of distance-regular graphs, namely the subgraph of the dual polar graph of type B_3(q) induced on the vertices far from a fixed point, and the subgraph of the dual polar graph of type D_4(q) induced on the vertices far from a fixed edge. The latter is the extended bipartite double of the former.

math.CO↗