We show that n-bit integers can be factorized by independently running a quantum circuit with \(\tilde{O}(n^{3/2}) \) gates for \(\sqrt {n}+4 \) times, and then using polynomial-time classical post-processing. The correctness of the algorithm relies ...

A new Las Vegas algorithm is presented for the composition of two polynomials modulo a third one, over an arbitrary field. When the degrees of these polynomials are bounded by n, the algorithm uses O(n^1.43) field operations, breaking through the 3/2 ...

A collection of sets displays a proximity gap with respect to some property if for every set in the collection, either (i) all members are δ-close to the property in relative Hamming distance or (ii) only a tiny fraction of members are δ-close to the ...

A polynomial program is one in which all assignments are given by polynomial expressions and in which all branching is nondeterministic (as opposed to conditional). Given such a program, an algebraic invariant is one that is defined by polynomial ...

We consider the problem of deciding the existence of real roots of real-valued exponential polynomials with algebraic coefficients. Such functions arise as solutions of linear differential equations with real algebraic coefficients. We focus on two ...

Newton iteration is an almost 350-year-old recursive formula that approximates a simple root of a polynomial quite rapidly. We generalize it to a matrix recurrence (allRootsNI) that approximates all roots simultaneously. In this form, the process yields ...

Assuming a widely believed hypothesis concerning the least prime in an arithmetic progression, we show that polynomials of degree less than \( n \) over a finite field \( \mathbb {F}_q \) with \( q \) elements can be multiplied in time \(\( O (n \log ...\)

Strassen’s algorithm (1969) was the first sub-cubic matrix multiplication algorithm. Winograd (1971) improved the leading coefficient of its complexity from 6 to 7. There have been many subsequent asymptotic improvements. Unfortunately, most of these ...

We describe and analyze an algorithm for computing the homology (Betti numbers and torsion coefficients) of basic semialgebraic sets that works in weak exponential time. That is, of a set of exponentially small measure in the space of data, the cost of ...

Monadic predicates play a prominent role in many decidable cases, including decision procedures for symbolic automata. We are here interested in discovering whether a formula can be rewritten into a Boolean combination of monadic predicates. Our setting ...

Polynomials over finite fields play a central role in algorithms for cryptography, error correcting codes, and computer algebra. The complexity of multiplying such polynomials is still a major open problem. Let p be a prime, and let M_p(n) denote the bit ...

Algebraic proof systems, such as Polynomial Calculus (PC) and Polynomial Calculus with Resolution (PCR), refute contradictions using polynomials. Space complexity for such systems measures the number of distinct monomials to be kept in memory while ...

In this article, we investigate the logical structure of memory models of theoretical and practical interest. Our main interest is in “the logic behind a fixed memory model”, rather than in “a model of any kind behind a given logical system”. As an ...

We prove that multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NP-hard. Our list includes: determining the feasibility of a system of bilinear equations, deciding whether a 3-tensor possesses a given ...

We study the problem of identity testing for depth-3 circuits of top fanin k and degree d. We give a new structure theorem for such identities that improves the known deterministic d^kO(k)-time blackbox identity test over rationals [Kayal and Saraf, 2009]...

We analyze the convergence of randomized trace estimators. Starting at 1989, several algorithms have been proposed for estimating the trace of a matrix by 1/MΣ_i=1^M z_i^T Az_i, where the z_i are random vectors; different estimators use different ...

Living organisms function in accordance with complex mechanisms that operate in different ways depending on conditions. Darwin's theory of evolution suggests that such mechanisms evolved through variation guided by natural selection. However, there has ...

Description logics (DLs) and rules are formalisms that emphasize different aspects of knowledge representation: whereas DLs are focused on specifying and reasoning about conceptual knowledge, rules are focused on nonmonotonic inference. Many ...

Abduction is a fundamental mode of reasoning with applications in many areas of AI and Computer Science. The computation of abductive explanations is an important computational problem, which is at the core of early systems such as the ATMS and Clause ...

We study random submatrices of a large matrix A. We show how to approximately compute A from its random submatrix of the smallest possible size O(rlog r) with a small error in the spectral norm, where r = ‖A‖²_F/‖A‖²₂ is the numerical rank of A. The ...