Related papers: Improved upper bound on root number of linearized …

We study the Excluded Grid Theorem of Robertson and Seymour. This is a fundamental result in graph theory, that states that there is some function $f: Z^+\rightarrow Z^+$, such that for all integers $g>0$, every graph of treewidth at least…

We study geometric characterizations of unbounded integer polynomial optimization problems. While unboundedness along a ray fully characterizes unbounded integer linear and quadratic optimization problems, we show that this is not the case…

We prove that, to compute a Boolean function $f$ on $N$ variables with error probability $\epsilon$, any quantum black-box algorithm has to query at least $\frac{1 - 2\sqrt{\epsilon}}{2} \rho_f N = \frac{1 - 2\sqrt{\epsilon}}{2} \bar{S}_f$…

Homology has long been accepted as an important computable tool for quantifying complex structures. In many applications, these structures arise as nodal domains of real-valued functions and are therefore amenable only to a numerical study…

Linearized polynomials have attracted a lot of attention because of their applications in both geometric and algebraic areas. Let $q$ be a prime power, $n$ be a positive integer and $\sigma$ be a generator of…

In this note we consider roots of multivariate polynomials over a finite grid. When given information on the leading monomial with respect to a fixed monomial ordering, the footprint bound [8, 5] provides us with an upper bound on the…

Let $f$ be the $\mathbb{F}_q$-linear map over $\mathbb{F}_{q^{2n}}$ defined by $x\mapsto x+ax^{q^s}+bx^{q^{n+s}}$ with $\gcd(n,s)=1$. It is known that the kernel of $f$ has dimension at most $2$, as proved by Csajb\'ok et al. in "A new…

This paper considers fast algorithms for operations on linearized polynomials. We propose a new multiplication algorithm for skew polynomials (a generalization of linearized polynomials) which has sub-quadratic complexity in the polynomial…

Motivated by practical applications, recent works have considered maximization of sums of a submodular function $g$ and a linear function $\ell$. Almost all such works, to date, studied only the special case of this problem in which $g$ is…

Understanding the query complexity for testing linear-invariant properties has been a central open problem in the study of algebraic property testing. Triangle-freeness in Boolean functions is a simple property whose testing complexity is…

We introduce a subexponential algorithm for geometric solving of multivariate polynomial equation systems whose bit complexity depends mainly on intrinsic geometric invariants of the solution set. From this algorithm, we derive a new…

The paper considers a global version of the notion of log canonical threshold for plurisubharmonic functions $u$ of logarithmic growth in $\mathbb{C}^n$, aiming at description of the range of all $p>0$ such that $e^{-u}\in…

The motivation of this work stems from the numerical approximation of bounded functions by polynomials satisfying the same bounds. The present contribution makes use of the recent algebraic characterization found in [B. Despr\'es, Numer.…

We investigate the max min of the algebraic degree and the nonlinearity of a Boolean function in $n$ variables when restricted to a $k$-dimensional affine subspace of $\mathbb{F}_2^n$. Previous authors have focused on the cases when the max…

We consider multivariable polynomials over a fixed number field, linear in some of the variables. For a system of such polynomials satisfying certain technical conditions we prove the existence of search bounds for simultaneous zeros with…

The factorization norms of the lower-triangular all-ones $n \times n$ matrix, $\gamma_2(M_{count})$ and $\gamma_{F}(M_{count})$, play a central role in differential privacy as they are used to give theoretical justification of the accuracy…

The absolute separation of a polynomial is the minimum nonzero difference between the absolute values of its roots. In the case of polynomials with integer coefficients, it can be bounded from below in terms of the degree and the height…

Universal approximation theorems show that neural networks can approximate any continuous function; however, the number of parameters may grow exponentially with the ambient dimension, so these results do not fully explain the practical…

We study the problem of computing the tightest upper and lower bounds on the probability that the sum of $n$ dependent Bernoulli random variables exceeds an integer $k$. Under knowledge of all pairs of bivariate distributions denoted by a…

Though it is well known that the roots of any affine polynomial over a finite field can be computed by a system of linear equations by using a normal base of the field, such solving approach appears to be difficult to apply when the field…