Searcharxiv⌕ Search

arXiv subjects

Fedor Petrov

Publications and source records attributed to Fedor Petrov.

At least 19 recordsLinked to original sources
arXiv2026

A greedoid and a matroid inspired by Bhargava's $p$-orderings

Consider a finite set $E$. Assume that each $e \in E$ has a "weight" $w \left(e\right) \in \mathbb{R}$ assigned to it, and any two distinct $e, f \in E$ have a "distance" $d \left(e, f\right) = d \left(f, e\right) \in \mathbb{R}$ assigned to them, such that the distances satisfy the ultrametric triangle inequality $d(a,b)\leqslant \max \left\{d(a,c),d(b,c)\right\}$. We look for a subset of $E$ of given size with maximum perimeter (where the perimeter is defined by summing the weights of all elements and their pairwise distances). We show that any such subset can be found by a greedy algorithm (which starts with the empty set, and then adds new elements one by one, maximizing the perimeter at each step). We use this to define numerical invariants, and also to show that the maximum-perimeter subsets of all sizes form a strong greedoid, and the maximum-perimeter subsets of any given size are the bases of a matroid. This essentially generalizes the "$P$-orderings" constructed by Bhargava in order to define his generalized factorials, and is also similar to the strong greedoid of maximum diversity subsets in phylogenetic trees studied by Moulton, Semple and Steel. We further discuss some numerical invariants of $E, w, d$ stemming from this construction, as well as an analogue where maximum-perimeter subsets are replaced by maximum-perimeter tuples (i.e., elements can appear multiple times).

math.CO↗
arXiv2026

Biregular bipartite labeled multigraphs and perfect matchings in bipartite tensor products

In 2019, P. Higgins formulated [1] a question about bipartite graphs (see Conjecture 1 below); this question arises in the study of regular finite semigroups. F. V. Petrov formulated [2] another combinatorial conjecture (Conjecture 3); Conjecture 3 implies Conjecture 1 and seems simple itself. However, both conjectures remain unproven in the general case. In the present paper, some special cases are proved, Conjecture 4 is formulated in the same spirit, and some of its special cases are proved. In addition, Conjecture 1 is reduced to a matrix inequality (Conjecture 5); this inequality is, in turn, also proved in a special case.

math.CO↗
arXiv2023

On the rank of the communication matrix for deterministic two-way finite automata

The communication matrix for two-way deterministic finite automata (2DFA) with $n$ states is defined for an automaton over a full alphabet of all $(2n+1)^n$ possible symbols: its rows and columns are indexed by strings, and the entry $(u, v)$ is $1$ if $uv$ is accepted by the automaton, and $0$ otherwise. With duplicate rows and columns removed, this is a square matrix of order $n(n^n-(n-1)^n)+1$, and its rank is known to be a lower bound on the number of states necessary to transform an $n$-state 2DFA to a one-way unambiguous finite automaton (UFA). This paper determines this rank, showing that it is exactly $f(n)=\sum_{k=1}^n \binom{n}{k-1} \binom{n}{k} \binom{2k-2}{k-1} =(1+o(1)) \frac{3\sqrt{3}}{8πn} 9^n$, and this function becomes the new lower bound on the state complexity of the 2DFA to UFA transformation, thus improving a recent lower bound by S. Petrov and Okhotin (``On the transformation of two-way deterministic finite automata to unambiguous finite automata'', Inf. Comput., 2023). The key element of the proof is determining the rank of a $k! \times k!$ submatrix, with its rows and columns indexed by permutations, where the entry $(π, σ)$ is $1$ if $σ\circ π$ is a cycle of length $k$, and 0 otherwise; using the methods of group representation theory it is shown that its rank is exactly $\binom{2k-2}{k-1}$, and this implies the above formula for $f(n)$.

cs.FL↗
arXiv2022

A combinatorial proof of the Burdzy-Pitman conjecture

We prove a sharp upper bound for the number of high degree differences in bipartite graphs: let $ (U, V, E)$ be a bipartite graph with $U=\{u_1, u_2, \dots, u_n\}$ and $V=\{v_1, v_2, \dots, v_n\}$; for $n\ge k>\frac{n}{2}$ we show that $\sum_{1\le i,j \le n} 1 {\Big\{|\text{deg}(u_i)-\text{deg}(v_j)|\ge k}\Big\} \le 2k(n-k).$ As a direct application we show a slightly stronger, probabilistic version of this theorem and thus confirm the Burdzy-Pitman conjecture about the maximal spread of coherent and independent distributions.

math.CO↗
arXiv2021

A generalization of Kruskal's theorem on tensor decomposition

Kruskal's theorem states that a sum of product tensors constitutes a unique tensor rank decomposition if the so-called k-ranks of the product tensors are large. We prove a "splitting theorem" for sets of product tensors, in which the k-rank condition of Kruskal's theorem is weakened to the standard notion of rank, and the conclusion of uniqueness is relaxed to the statement that the set of product tensors splits (i.e. is disconnected as a matroid). Our splitting theorem implies a generalization of Kruskal's theorem. While several extensions of Kruskal's theorem are already present in the literature, all of these use Kruskal's original permutation lemma, and hence still cannot certify uniqueness when the k-ranks are below a certain threshold. Our generalization uses a completely new proof technique, contains many of these extensions, and can certify uniqueness below this threshold. We obtain several other useful results on tensor decompositions as consequences of our splitting theorem. We prove sharp lower bounds on tensor rank and Waring rank, which extend Sylvester's matrix rank inequality to tensors. We also prove novel uniqueness results for non-rank tensor decompositions.

math.CO↗
arXiv2021

Alon -- Tarsi numbers of direct products

We provide a general framework on the coefficients of the graph polynomials of graphs which are Cartesian products. As a corollary, we prove that if $G=(V,E)$ is a graph with degrees of vertices $2d(v), v\in V$, and the graph polynomial $\prod_{(i,j)\in E} (x_j-x_i)$ contains an "almost central" monomial (that means a monomial $\prod_v x_v^{c_v}$, where $|c_v-d(v)|\leqslant 1$ for all $v\in V$), then the Cartesian product $G\square C_{2n}$ is $(d(\cdot)+2)$-choosable.

math.CO↗
arXiv2020

On the size of $A+λA$ for algebraic $λ$

For a finite set $A\subset \mathbb{R}$ and real $λ$, let $A+λA:=\{a+λb :\, a,b\in A\}$. Combining a structural theorem of Freiman on sets with small doubling constants together with a discrete analogue of Prékopa--Leindler inequality we prove a lower bound $|A+\sqrt{2} A|\geq (1+\sqrt{2})^2|A|-O({|A|}^{1-\varepsilon})$ which is essentially tight. We also formulate a conjecture about the value of $\liminf |A+λA|/|A|$ for an arbitrary algebraic $λ$. Finally, we prove a tight lower bound on the Lebesgue measure of $K+\mathcal{T} K$ for a given linear operator $\mathcal{T}\in \operatorname{End}(\mathbb{R}^d)$ and a compact set $K\subset \mathbb{R}^d$ with fixed measure. This continuous result supports the conjecture and yields an upper bound in it.

math.CO↗
arXiv2020

Hidden symmetries of weighted lozenge tilings

We study the weighted partition function for lozenge tilings, with weights given by multivariate rational functions originally defined by Morales, Pak and Panova (2019) in the context of the factorial Schur functions. We prove that this partition function is symmetric for large families of regions. We employ both combinatorial and algebraic proofs.

math.CO↗
arXiv2020

Combinatorial results implied by many zero divisors in a group ring

It has been recently proved (by Croot, Lev and Pach and the subsequent work by Ellenberg and Gijswijt) that for a group $G=G_0^n$, where $G_0\ne \{1,-1\}^m$ is a fixed finite Abelian group and $n$ is large, any subset $A$ without 3-progressions (triples $x,y,z$ of different elements with $xy=z^2$) contains at most $|G|^{1-c}$ elements, where $c>0$ is a constant depending only on $G_0$. This is known to be false when $G$ is, say, large cyclic group. The aim of this note is to show that algebraic property which corresponds to this difference is the following: in the first case a group algebra $\mathbb{F}[G]$ over suitable field $\mathbb{F}$ contains a subspace $X$ with codimension at most $|X|^{1-c}$ such that $X^3=0$. We discuss which bounds are obtained for finite Abelian $p$-groups and for some matrix $p$-groups: Heisenberg group over $\mathbb{F}_p$ and the unitriangular group over $\mathbb{F}_p$. Also we show how the method works for further generalizations by Kleinberg--Sawin--Speyer and Ellenberg.

math.CO↗
arXiv2020

On some determinants involving Jacobi symbols

In this paper we study some conjectures on determinants with Jacobi symbol entries posed by Z.-W. Sun. For any positive integer $n\equiv3\pmod4$, we show that $$(6,1)_n=[6,1]_n=(3,2)_n=[3,2]_n=0$$ and $$(4,2)_n=(8,8)_n=(3,3)_n=(21,112)_n=0$$ as conjectured by Sun, where $$(c,d)_n=\bigg|\left(\frac{i^2+cij+dj^2}n\right)\bigg|_{1\le i,j\le n-1}$$ and $$[c,d]_n=\bigg|\left(\frac{i^2+cij+dj^2}n\right)\bigg|_{0\le i,j\le n-1}$$ with $(\frac{\cdot}n)$ the Jacobi symbol. We also prove that $(10,9)_p=0$ for any prime $p\equiv5\pmod{12}$, and $[5,5]_p=0$ for any prime $p\equiv 13,17\pmod{20}$, which were also conjectured by Sun. Our proofs involve character sums over finite fields.

math.NT↗
arXiv2020

Proof of some conjectures involving quadratic residues

We confirm several conjectures of Sun involving quadratic residues modulo odd primes. For any prime $p\equiv 1\pmod 4$ and integer $a\not\equiv0\pmod p$, we prove that \begin{align*}&(-1)^{|\{1\le k<\frac p4:\ (\frac kp)=-1\}|}\prod_{1\le j \{ak^2\}_p\right\}\right| \\&+\left|\left\{(j,k):\ 1\le j \frac p2\right\}\right| \\\equiv&\left|\left\{1\le k<\frac p4:\ \left(\frac kp\right)=\left(\frac ap\right)\right\}\right|\pmod2. \end{align*} where $(\frac{a}p)$ is the Legendre symbol, $\varepsilon_p$ and $h(p)$ are the fundamental unit and the class number of the real quadratic field $\mathbb Q(\sqrt p)$ respectively, and $\{x\}_p$ is the least nonnegative residue of an integer $x$ modulo $p$. Also, for any prime $p\equiv3\pmod4$ and $δ=1,2$, we determine $$(-1)^{\left|\left\{(j,k): \ 1\le j \{δT_k\}_p\right\}\right|},$$ where $T_m$ denotes the triangular number $m(m+1)/2$.

math.NT↗
arXiv2019

Bang's problem and symplectic invariants

We consider the Tarski--Bang problem about covering of convex bodies by planks. The results of this kind give a lower bound on the sum of widths of planks (regions between a pair of parallel hyperplanes) covering a given convex body. Previously we have applied some notions of symplectic geometry to study convex bodies, and here we show that the symplectic techniques may be useful in this problem as well. We are able to handle some particular cases with the symplectic techniques, and show that the general cases would follow from a certain ``subadditivity conjecture'' in symplectic geometry, motivated by the results of K.~Ball. We also prove several related results by more elementary methods.

math.MG↗
arXiv2019

The Alon-Tarsi Number of A Toroidal Grid

The Alon-Tarsi number $AT(G)$ of a graph $G$ is the smallest $k$ for which there is an orientation $D$ of $G$ with max indegree $k-1$ such that the number of even and odd circulations contained in D are different. In this paper, we show that the Alon--Tarsi number of toroidal grids $T_{m,n}=C_m\Box C_n$ equals $4$ when $m,n$ are both odd and $3$ otherwise.

math.CO↗
arXiv2019

A remark on sets with few distances in $\mathbb{R}^{d}$

A celebrated theorem due to Bannai-Bannai-Stanton says that if $A$ is a set of points in $\mathbb{R}^{d}$, which determines $s$ distinct distances, then $$|A| \leq {d+s \choose s}.$$ In this note, we give a new simple proof of this result by combining Sylvester's Law of Inertia for quadratic forms with the proof of the so-called Croot-Lev-Pach Lemma from additive combinatorics.

math.CO↗
arXiv2019

Regular behaviour of the maximal hypergraph chromatic number

Let $m(n,r)$ denote the minimal number of edges in an $n$-uniform hypergraph which is not $r$-colorable. It is known that for a fixed $n$ one has \[ c_n r^n < m(n,r) < C_n r^n. \] We prove that for any fixed $n$ the sequence $a_r := m(n,r)/r^n$ has a limit, which was conjectured by Alon. We also prove the list colorings analogue of this statement.

math.CO↗