Related papers: True complexity and iterated Cauchy--Schwarz
According to the real \tau-conjecture, the number of real roots of a sum of products of sparse polynomials should be polynomially bounded in the size of such an expression. It is known that this conjecture implies a superpolynomial lower…
We prove an identity for sesquilinear maps from the Cartesian square of a vector space to a geometric mean closed Archimedean (real or complex) vector lattice, from which the Cauchy-Schwarz inequality follows. A reformulation of this result…
In this article,we first give a modified Schwarz-Pompeiu formula in a general sector ring by proper conformal mappings, and obtain the solution of the Schwarz problem for the Cauchy-Riemann equation in explicit forms. Furthermore, a class…
The field of computational complexity is concerned both with the intrinsic hardness of computational problems and with the efficiency of algorithms to solve them. Given such a problem, normally one designs an algorithm to solve it and sets…
According to Kumar's recent surprising result (ToCT'20), a small border Waring rank implies that the polynomial can be approximated as a sum of a constant and a small product of linear polynomials. We prove the converse of Kumar's result…
Reasoning about real number expressions in a proof assistant is challenging. Several problems in theorem proving can be solved by using exact real number computation. I have implemented a library for reasoning and computing with complete…
The graph isomorphism problem looks deceptively simple, but although polynomial-time algorithms exist for certain types of graphs such as planar graphs and graphs with bounded degree or eigenvalue multiplicity, its complexity class is still…
We prove tight upper bounds for the number of vertices of a simple polygon that is the union or the intersection of two simple polygons with given numbers of convex and concave vertices. The similar question on graphs of the lower (or…
We consider a subclass of tilings, the tilings obtained by cut and projection. Under somewhat standard assumptions, we show that the natural complexity function has polynomial growth. We compute its exponent \alpha in terms of the ranks of…
In this article we call a sequence $(a_n)_n$ of elements of a metric space nearly computably Cauchy if for every strictly increasing computable function $r:\mathbb{N}\to\mathbb{N}$ the sequence $(d(a_{r(n+1)},a_{r(n)}))_n$ converges…
We extend the Ax-Katz theorem for a single polynomial from finite fields to the rings Z_m with m composite. This extension not only yields the analogous result, but gives significantly higher divisibility bounds. We conjecture what computer…
We derive analytic solutions for the potential and field in a one-dimensional system of masses or charges with periodic boundary conditions, in other words Ewald sums for one dimension. We also provide a set of tools for exploring the…
In this article, we establish an improvement of the Cauchy-Schwarz inequality. Let $x, y \in \mathcal{H},$ and let $f: (0,1) \rightarrow \mathbb{R}^+$ be a well-defined function, where $\mathbb{R}^+$ denote the set of all positive real…
The rational covariance extension problem to determine a rational spectral density given a finite number of covariance lags can be seen as a matrix completion problem to construct an infinite-dimensional positive-definite Toeplitz matrix…
The main goal of this paper is to show that if a real valued function defined on a groupoid satisfies a certain Levi--Civita-type functional equation, then it also fulfills a Cauchy--Schwarz-type functional inequality. In particular, if the…
We show that most arithmetic circuit lower bounds and relations between lower bounds naturally fit into the representation-theoretic framework suggested by geometric complexity theory (GCT), including: the partial derivatives technique…
Given a hyper loop algebra over a non-algebraically closed field, we address multiplicity problems in the underlying abelian tensor category of finite-dimensional representations. Namely, we give formulas for the l-characters of the simple…
We introduce the notion of being Weihrauch-complete for layerwise computability and provide several natural examples related to complex oscillations, the law of the iterated logarithm and Birkhoff's theorem. We also consider hitting time…
In the algebraic metacomplexity framework we prove that the decomposition of metapolynomials into their isotypic components can be implemented efficiently, namely with only a quasipolynomial blowup in the circuit size. We use this to…
We prove an effective version of the inverse theorem for the Gowers $U^3$-norm for functions supported on high-rank quadratic level sets in finite vector spaces. For configurations controlled by the $U^3$-norm (complexity-two…