Related papers: Cantor polynomials and the Fueter-Polya theorem

The problem of writing real zero polynomials as determinants of linear matrix polynomials has recently attracted a lot of attention. Helton and Vinnikov have proved that any real zero polynomial in two variables has a determinantal…

We show that the only orthogonal polynomials with a generating function of the form $F(x z - \alpha z^2)$ are the ultraspherical, Hermite, and Chebyshev polynomials of the first kind. For special $F$ for which this is the case, we then…

Zernike polynomials are a basis of orthogonal polynomials on the unit disk that are a natural basis for representing smooth functions. They arise in a number of applications including optics and atmospheric sciences. In this paper, we…

We prove that a self-similar Cantor set in $\mathbb{Z}_N \times \mathbb{Z}_N$ has a fractal uncertainty principle if and only if it does not contain a pair of orthogonal lines. The key ingredient in our proof is a quantitative form of…

A polynomial has saturated Newton polytope (SNP) if every lattice point of the convex hull of its exponent vectors corresponds to a monomial. We compile instances of SNP in algebraic combinatorics (some with proofs, others conjecturally):…

Let $F$ be a number field. Given a quadratic polynomial $f_c(z) = z^2 + c \in F[z]$, we can construct a directed graph $Preper(f_c, F)$ (also called a portrait), whose vertices are $F$-rational preperiodic points for $f_c$, with an edge…

We prove that the cluster monomials in non-initial cluster variables are uniquely determined by the Newton polytopes of their $F$-polynomials for skew-symmetrizable cluster algebras. Accordingly, we prove that the $\tau$-rigid modules and…

A partition is finitary if all its members are finite. For a set $A$, $\mathscr{B}(A)$ denotes the set of all finitary partitions of $A$. It is shown consistent with $\mathsf{ZF}$ (without the axiom of choice) that there exist an infinite…

If $\mathcal{P}$ is a lattice polytope (i.e., $\mathcal{P}$ is the convex hull of finitely many integer points in $\mathbb{R}^d$), Ehrhart's famous theorem (1962) asserts that the integer-point counting function $|t \mathcal{P} \cap…

Consider the problem, usually called the P\'olya-Chebotarev problem, of finding a continuum in the complex plane including some given points such that the logarithmic capacity of this continuum is minimal. We prove that each connected…

A wide variety of problems in combinatorics and discrete optimization depend on counting the set $S$ of integer points in a polytope, or in some more general object constructed via discrete geometry and first-order logic. We take a tour…

An invertible polynomial in $n$ variables is a quasihomogeneous polynomial consisting of $n$ monomials so that the weights of the variables and the quasi-degree are well defined. In the framework of the construction of mirror symmetric…

A theorem of Scott gives an upper bound for the normalized volume of lattice polygons with exactly $i>0$ interior lattice points. We will show that the same bound is true for the normalized volume of lattice polytopes of degree 2 even in…

A non-zero constant Jacobian polynomial maps $F=(P,Q)$ of $\mathbb{C}^2$ is invertible if $P$ and $Q$ are rational polynomials.

By means of a nested sequence of some critical pieces constructed by Kozlovski, Shen, and van Strien, and by using a covering lemma recently proved by Kahn and Lyubich, we prove that the Julia set of a polynomial is a Cantor set if and only…

We prove a quantitative Roth-type theorem for polynomial corners in $\mathbb{R}^2$. Let $P_1$ and $P_2$ be two linearly independent polynomials with zero constant term. We show that any measurable subset of $[0,1]^2$ with positive measure…

Let $K$ be the closure of a bounded region in the complex plane with simply connected complement whose boundary is a piecewise analytic curve with at least one outward cusp. The asymptotics of zeros of Faber polynomials for $K$ are not…

In this work we present a new polynomial map $f:=(f_1,f_2):{\mathbb R}^2\to{\mathbb R}^2$ whose image is the open quadrant $\{x>0,y>0\}\subset{\mathbb R}^2$. The proof of this fact involves arguments of topological nature that avoid hard…

In this paper we show that a split central simple algebra with quadratic pair which decomposes into a tensor product of quaternion algebras with involution and a quaternion algebra with quadratic pair is adjoint to a quadratic Pfister form.…

For any polynomial $P \in \mathbb{C}[X_1,X_2,...,X_n]$, we describe a $\mathbb{C}$-vector space $F(P)$ of solutions of a linear system of equations coming from some algebraic partial differential equations such that the dimension of $F(P)$…