Related papers: A Quantitative Doignon-Bell-Scarf Theorem

We give a strongly polynomial-time algorithm for integer linear programs defined by integer coefficient matrices whose subdeterminants are bounded by a constant and that contain at most two nonzero entries in each row. The core of our…

Let K be a complete discretely valued field with residue field k. If char(K) = 0, char(k) = 2 and the 2-rank of k is d, we prove that there exists an integer N depending on d such that the u-invariant of any function field in one variable…

It is a recent realization that many of the concepts and tools of causal discovery in machine learning are highly relevant to problems in quantum information, in particular quantum nonlocality. The crucial ingredient in the connection…

Let $\sigma_i$, $i=1,\ldots,n$, denote positive Borel measures on $\mathbb{R}^d$, let $\mathcal{D}$ denote the usual collection of dyadic cubes in $\mathbb{R}^d$ and let $K:\,\mathcal{D}\to[0,\infty)$ be a~map. In this paper we give…

Let $A \in \mathbb{Z}^{m \times n}$ be an integer matrix with components bounded by $\Delta$ in absolute value. Cook et al.~(1986) have shown that there exists a universal matrix $B \in \mathbb{Z}^{m' \times n}$ with the following property:…

Carleson's Theorem asserts the pointwise convergence of Fourier series of square integrable functions. We give a complete proof, following joint work of the author and C. Thiele. Over 20 exercises are also detailed. We also discuss the…

We study the copolynomials of $n$ variables, i.e. $K$-linear mappings from the ring of polynomials $K[x_1,...,x_n]$ into the commutative ring $K$. We prove an existence and uniqueness theorem for a linear differential equation of infinite…

Using the circle method in combination with lattice point counting arguments, we show that for almost all homogeneous diophantine equations of additive type and degree $k$ in more than $4k$ variables, the Local-Global principle holds true.…

Let $k$ be a field of characteristic zero and ${\bar k}$ an algebraic closure of $k$. For a geometrically integral variety $X$ over $k$, we write ${\bar k}(X)$ for the function field of ${\bar X}=X\times_k{\bar k}$. If $X$ has a smooth…

We develop a quantitative theory of stochastic homogenization for linear, uniformly parabolic equations with coefficients depending on space and time. Inspired by recent works in the elliptic setting, our analysis is focused on certain…

In this paper we characterize real bivariate polynomials which have a small range over large Cartesian products. We show that for every constant-degree bivariate real polynomial $f$, either $|f(A,B)|=\Omega(n^{4/3})$, for every pair of…

We study the mixed Christoffel problem for $C^{2,+}$ convex bodies providing sufficient conditions for its solution. Key to our approach is a constant rank theorem, following the approach developed in \cite{Guan-Ma-2003} to address the…

Constraint satisfaction (CSP) and structure isomorphism (SI) are among the most well-studied computational problems in Computer Science. While neither problem is thought to be in $\texttt{PTIME},$ much work is done on $\texttt{PTIME}$…

We extend the Kahane-Katznelson-de Leeuw theorem to smoothness spaces by showing that for any $g \in W^{l,2}(\mathbb{T}^d)$, there exists a function $f\in C^l(\mathbb{T}^d)$ satisfying $|\widehat{f}(n)|\geq |\widehat{g}(n)|$ and…

The disjointness of a stabilizer code is a quantity used to constrain the level of the logical Clifford hierarchy attainable by transversal gates and constant-depth quantum circuits. We show that for any positive integer constant $c$, the…

We consider a stochastic partial differential equation with logarithmic (or negative power) nonlinearity, with one reflection at 0 and with a constraint of conservation of the space average. The equation, driven by the derivative in space…

It is well known that the Fourier--Bohr coefficients of regular model sets exist and are uniformly converging, volume-averaged exponential sums. Several proofs for this statement are known, all of which use fairly abstract machinery. For…

The Cauchy problem for the classical Dirac-Klein-Gordon system in two space dimensions is globally well-posed for L^2 Schoedinger data and wave data in H^{1/2} \times H^{-1/2}. In the case of smooth data there exists a global smooth…

Assume that $k \le d$ is a positive integer and $\C$ is a finite collection of convex bodies in $\R^d$. We prove a Helly type theorem: If for every subfamily $\C^*\subset \C$ of size at most $\max \{d+1,2(d-k+1)\}$ the set $\bigcap \C^*$…

We provide the final step in the resolution of Bourgain's slicing problem in the affirmative. Thus we establish the following theorem: for any convex body $K \subseteq \mathbb{R}^n$ of volume one, there exists a hyperplane $H \subseteq…