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Bogdan Nica

Publications and source records attributed to Bogdan Nica.

At least 19 recordsLinked to original sources
arXiv2026

Counting in Vieta graphs over $\mathbb{F}_p$

We introduce and study a finite simple graph of algebraic origin: the Vieta graph on the solution set over $\mathbb{F}_p$ to a symmetric, multivariate equation which is quadratic in each variable. This construction is a broad generalization of the Markoff graph over $\mathbb{F}_p$, extensively studied in the recent literature. We give a systematic approach, partly based on quadratic character sums, to the following basic counting questions: how many vertices does a Vieta graph have, and what is the degree distribution? We focus on explicit counts, addressing the low-dimensional cases in three and four variables.

math.NT↗
arXiv2024

Norms of spherical averaging operators for some geometric group actions

We obtain asymptotic estimates for the $\ell^p$-operator norm of spherical averaging operators associated to certain geometric group actions. The motivating example is the case of Gromov hyperbolic groups, for which we obtain asymptotically sharp estimates. We deduce asymptotic lower bounds for the combinatorial expansion of spheres.

math.GR↗
arXiv2024

On the independence number of regular graphs of matrix rings

Consider a graph on the non-singular matrices over a finite field, in which two distinct non-singular matrices are joined by an edge whenever their sum is singular. We prove an upper bound for the independence number of this graph. As a consequence, we obtain a lower bound for its chromatic number that significantly improves a previous result of Tomon.

math.CO↗
arXiv2023

On an identity of Sylvester

We discuss an algebraic identity, due to Sylvester, as well as related algebraic identities and applications.

math.CO↗
arXiv2021

Polynomials over structured grids

We study multivariate polynomials over `structured' grids. We begin by proposing an interpretation as to what it means for a finite subset of a field to be structured; we do so by means of a numerical parameter, the nullity. We then extend several results--notably, the Combinatorial Nullstellensatz and the Coefficient Theorem--to polynomials over structured grids. The main point is that the structure of a grid allows the degree constraints on polynomials to be relaxed.

math.CO↗
arXiv2019

Kloosterman sums over finite Frobenius rings

We study Kloosterman sums in a generalized ring-theoretic context, that of finite commutative Frobenius rings. We prove a number of identities for twisted Kloosterman sums, loosely clustered around moment computations.

math.NT↗
arXiv2019

A relative bound for independence

We prove an upper bound for the independence number of a graph in terms of the largest Laplacian eigenvalue, and of a certain induced subgraph. Our bound is a refinement of a well-known Hoffman-type bound.

math.CO↗
arXiv2019

Two applications of strong hyperbolicity

We present two analytic applications of the fact that a hyperbolic group can be endowed with a strongly hyperbolic metric. The first application concerns the crossed-product C*-algebra defined by the action of a hyperbolic group on its boundary. We construct a natural time flow, involving the Busemann cocycle on the boundary. This flow has a natural KMS state, coming from the Hausdorff measure on the boundary, which is furthermore unique when the group is torsion-free. The second application is a short new proof of the fact that a hyperbolic group admits a proper isometric action on an $\ell^p$-space, for large enough $p$.

math.GR↗
arXiv2016

Unimodular graphs and Eisenstein sums

Motivated in part by combinatorial applications to certain sum-product phenomena, we introduce unimodular graphs over finite fields and, more generally, over finite valuation rings. We compute the spectrum of the unimodular graphs, by using Eisenstein sums associated to unramified extensions of such rings. We derive an estimate for the number of solutions to the restricted dot product equation $a\cdot b=r$ over a finite valuation ring. Furthermore, our spectral analysis leads to the exact value of the isoperimetric constant for half of the unimodular graphs. We also compute the spectrum of Platonic graphs over finite valuation rings, and products of such rings - e.g., $\mathbb{Z}/(N)$. In particular, we deduce an improved lower bound for the isoperimetric constant of the Platonic graph over $\mathbb{Z}/(N)$.

math.CO↗
arXiv2015

K-homological finiteness and hyperbolic groups

Motivated by classical facts concerning closed manifolds, we introduce a strong finiteness property in K-homology. We say that a C*-algebra has uniformly summable K-homology if all its K-homology classes can be represented by Fredholm modules which are finitely summable over the same dense subalgebra, and with the same degree of summability. We show that two types of C*-algebras associated to hyperbolic groups - the C*-crossed product for the boundary action, and the reduced group C*-algebra - have uniformly summable K-homology. We provide explicit summability degrees, as well as explicit finitely summable representatives for the K-homology classes.

math.OA↗
arXiv2015

On operator norms for hyperbolic groups

We estimate the operator norm of radial non-negative functions on hyperbolic groups. As a consequence, we show that several forms of Haagerup's inequality are optimal.

math.GR↗
arXiv2015

Strong hyperbolicity

We propose the metric notion of strong hyperbolicity as a way of obtaining hyperbolicity with sharp additional properties. Specifically, strongly hyperbolic spaces are Gromov hyperbolic spaces that are metrically well-behaved at infinity, and, under weak geodesic assumptions, they are strongly bolic as well. We show that CAT(-1) spaces are strongly hyperbolic. On the way, we determine the best constant of hyperbolicity for the standard hyperbolic plane. We also show that the Green metric defined by a random walk on a hyperbolic group is strongly hyperbolic. A measure-theoretic consequence at the boundary is that the harmonic measure defined by a random walk is a visual Hausdorff measure.

math.GR↗
arXiv2014

A true relative of Suslin's normality theorem

We prove a normality theorem for the "true" elementary subgroups of $SL_n(A)$ defined by the ideals of a commutative unital ring $A$. Our result is an analogue of a normality theorem, due to Suslin, for the standard elementary subgroups, and it greatly generalizes a theorem of Mennicke.

math.GR↗
arXiv2013

The Mazur-Ulam theorem

A short proof of the Mazur-Ulam theorem concerning isometries of real normed spaces.

math.MG↗