Related papers: Algebraic Pseudorandomness in $VNC^0$

This paper introduces Roundabout Constrained Convex Generators (RCGs), a set representation framework for modeling multiply connected regions in control and verification applications. The RCG representation extends the constrained convex…

We establish connections between state tomography, pseudorandomness, quantum state synthesis, and circuit lower bounds. In particular, let $\mathfrak{C}$ be a family of non-uniform quantum circuits of polynomial size and suppose that there…

We establish the notion of a ``projective analytic vector'', whose defining requirements are weaker than the usual ones of an analytic vector, and use it to prove generation theorems for one-parameter groups on locally convex spaces. More…

We deploy algebraic complexity theoretic techniques for constructing symmetric determinantal representations of for00504925mulas and weakly skew circuits. Our representations produce matrices of much smaller dimensions than those given in…

We establish the first general connection between the design of quantum algorithms and circuit lower bounds. Specifically, let $\mathfrak{C}$ be a class of polynomial-size concepts, and suppose that $\mathfrak{C}$ can be PAC-learned with…

This work initiates the systematic study of explicit distributions that are indistinguishable from a single exponential-size combinatorial object. In this we extend the work of Goldreich, Goldwasser and Nussboim (SICOMP 2010) that focused…

While efficient randomized algorithms for factorization of polynomials given by algebraic circuits have been known for decades, obtaining an even slightly non-trivial deterministic algorithm for this problem has remained an open question of…

The Ideal Proof System (IPS) of Grochow & Pitassi (FOCS 2014, J. ACM, 2018) is an algebraic proof system that uses algebraic circuits to refute the solvability of unsatisfiable systems of polynomial equations. One potential drawback of IPS…

We study a geometric hitting-set problem in which the input consists of a set $P$ of weighted points and a family $S=H\cup V$ of axis-parallel segments in the plane. The goal is to select a minimum-weight subset of $P$ that hits every…

This is a study of the computing power of the subtyping machine behind Kennedy and Pierce's nominal subtyping with variance. We depict the lattice of fragments of Kennedy and Pierce's type system and characterize their computing power in…

We show a new PRG construction fooling depth-$d$, size-$m$ $\mathsf{AC}^0$ circuits within error $\varepsilon$, which has seed length $O(\log^{d-1}(m)\log(m/\varepsilon)\log\log(m))$. Our PRG improves on previous work (Trevisan and Xue…

Lie Algebras are powerful mathematical structures used in physics to describe sets of operators and associated combinations. A central task is to identify a minimal set of generators from which the algebra can be constructed. The classical…

I discuss a seemingly unlikely confluence of topics in algebra, numerical computation, and computer vision. The motivating problem is that of solving multiples instances of a parametric family of systems of algebraic (polynomial or rational…

Random numbers are central to cryptography and various other tasks. The intrinsic probabilistic nature of quantum mechanics has allowed us to construct a large number of quantum random number generators (QRNGs) that are distinct from the…

We study pseudorandomness and pseudorandom generators from the perspective of logical definability. Building on results from ordinary derandomization and finite model theory, we show that it is possible to deterministically construct, in…

We investigate the geometric hitting set problem in the online setup for the range space $\Sigma=({\cal P},{\cal S})$, where the set $\P\subset\mathbb{R}^2$ is a collection of $n$ points and the set $\cal S$ is a family of geometric objects…

Let $P_k:= \mathbb F_2[x_1,x_2,\ldots,x_k]$ be the polynomial algebra over the prime field of two elements, $\mathbb F_2$, in $k$ variables $x_1, x_2, \ldots, x_k$, each of degree 1. We are interested in the Peterson hit problem of finding…

In this work, we construct explicit formulas for the generators of the Cartan centralisers of complex semisimple Lie algebras $B_n,C_n$ and $D_n$, the case $A_n$ being already known \cite{campoamor2023algebraic}. The precise structures for…

Gaussian random number generators attract a widespread interest due to their applications in several fields. Important requirements include easy implementation, tail accuracy, and, finally, a flat spectrum. In this work, we study the…

Let G be a unipotent algebraic subgroup of some GL_m(C) defined over Q. We describe an algorithm for finding a finite set of generators of the subgroup G(Z) = G \cap GL_m(Z). This is based on a new proof of the result (in more general form…