Related papers: Extending weakly polynomial functions from high ra…

The vector space of all polynomial functions of degree $k$ on a box of dimension $n$ is of dimension ${n \choose k}$. A consequence of this fact is that a function can be approximated on vertices of the box using other vertices to higher…

Let $[n]$ (resp. $V$) be an $n$-element set (resp. $n$-dimensional vector space over the finite field $\mathbb{F}_{q}$), and $\binom{[n]}{k}$ (resp. $\genfrac{[}{]}{0pt}{}{V}{k}$) denote the set of all $k$-subsets of $[n]$ (resp.…

We obtain upper bounds, independent of the ambient dimension, for the number of realizable zero-nonzero patterns and (over ordered fields) sign conditions of a finite family of polynomials $\mathcal P$ restricted to an algebraic subset $V$…

Let $\mathcal{G}$ be a minor-closed graph class. We say that a graph $G$ is a $k$-apex of $\mathcal{G}$ if $G$ contains a set $S$ of at most $k$ vertices such that $G\setminus S$ belongs to $\mathcal{G}.$ We denote by $\mathcal{A}_k…

In this paper we investigate the uniform distribution properties of polynomials in many variables and bounded degree over a fixed finite field F of prime order. Our main result is that a polynomial P : F^n -> F is poorly-distributed only if…

In the large rank limit, for any nonexceptional affine algebra, the graded branching multiplicities known as one-dimensional sums, are conjectured to have a simple relationship with those of type A, which are known as generalized Kostka…

A weakly complete space is a complex space admitting a (smooth) plurisubharmonic exhaustion function. In this paper, we classify those weakly complete complex surfaces for which such exhaustion function can be chosen real analytic: they can…

Consider a Henselian rank one valued field $K$ of equicharacteristic zero along with the language $\mathcal{L}^{P}$ of Denef--Pas. Let $f: A \to K$ be an $\mathcal{L}^{P}$-definable (with parameters) function on a subset $A$ of $K^{n}$. We…

A regularity lemma for polynomials provides a decomposition in terms of a bounded number of approximately independent polynomials. Such regularity lemmas play an important role in numerous results, yet suffer from the familiar shortcoming…

Let K be F_q((T)), or more generally any field of characteristic p equipped with a valuation having a finite residue field of q elements. Then a polynomial f(x) in K[x] having k+1 nonzero coefficients has at most q^k distinct zeros in K. We…

Let M be a smooth manifold, and let O(M) be the poset of open subsets of M. Manifold calculus, due to Goodwillie and Weiss, is a calculus of functors suitable for studying contravariant functors (cofunctors) F: O(M)--> Top from O(M) to the…

The problem of constructing explicit functions which cannot be approximated by low degree polynomials has been extensively studied in computational complexity, motivated by applications in circuit lower bounds, pseudo-randomness,…

In this paper we present an unexpected link between the Factorial Conjecture and Furter's Rigidity Conjecture. The Factorial Conjecture in dimension $m$ asserts that if a polynomial $f$ in $m$ variables $X_i$ over $\C$ is such that ${\cal…

A slice decomposition is an expression of a homogeneous polynomial as a sum of forms with a linear factor. A strength decomposition is an expression of a homogeneous polynomial as a sum of reducible forms. The slice rank and strength of a…

We consider convex sets and functions over idempotent semifields, like the max-plus semifield. We show that if $K$ is a conditionally complete idempotent semifield, with completion $\bar{K}$, a convex function $K^n\to\bar{K}$ which is lower…

Given $d,n \in \mathbb{N}$, we write a polynomial $F \in \mathbb{C}[x_1,\dots,x_n]$ to be degenerate if there exist $P\in \mathbb{C}[y_1, \dots, y_{n-1}]$ and $m_j = x_1^{v_{j,1}}\dots x_n^{v_{j,n}}$ with $v_{j,1}, \dots, v_{j,n} \in…

Given a closed subvariety X in a projective space, the rank with respect to X of a point p in this projective space is the least integer r such that p lies in the linear span of some r points of X. Let W_k be the closure of the set of…

We prove a function field analogue of Maynard's result about primes with restricted digits. That is, for certain ranges of parameters n and q, we prove an asymptotic formula for the number of irreducible polynomials of degree n over a…

Let F = F_p for any fixed prime p >= 2. An affine-invariant property is a property of functions on F^n that is closed under taking affine transformations of the domain. We prove that all affine-invariant property having local…

We study irreducible subvarieties of the universal hypersurface $\mathcal{X}/B$ of degree $d$ and dimension $n$. We prove that when $d$ is sufficiently large, a degree $kd$ subvariety $Z$ which dominates $B$ comes from intersection with a…