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Symbolic Mathematical tasks such as integration often require multiple well-defined steps and understanding of sub-tasks to reach a solution. To understand Transformers' abilities in such tasks in a fine-grained manner, we deviate from…
In the last years many results in the area of semidefinite programming were obtained for invariant (finite dimensional, or infinite dimensional) semidefinite programs - SDPs which have symmetry. This was done for a variety of problems and…
Template abstract domains allow to express more interesting properties than classical abstract domains. However, template generation is a challenging problem when one uses template abstract domains for program analysis. In this paper, we…
We consider the problem of approximating all real roots of a square-free polynomial $f$. Given isolating intervals, our algorithm refines each of them to a width of $2^{-L}$ or less, that is, each of the roots is approximated to $L$ bits…
We present a new algorithm to decompose generic spinor polynomials into linear factors. Spinor polynomials are certain polynomials with coefficients in the geometric algebra of dimension three that parametrize rational conformal motions.…
Methods for the computation of invariants and symmetries of nonlinear evolution, wave, and lattice equations are presented. The algorithms are based on dimensional analysis, and can be implemented in any symbolic language, such as…
We produce algorithms to detect whether a complex affine variety computed and presented numerically by the machinery of numerical algebraic geometry corresponds to an associated component of a polynomial ideal.
We review the method of symplectic invariants recently introduced to solve matrix models loop equations, and further extended beyond the context of matrix models. For any given spectral curve, one defined a sequence of differential forms,…
We introduce a subexponential algorithm for geometric solving of multivariate polynomial equation systems whose bit complexity depends mainly on intrinsic geometric invariants of the solution set. From this algorithm, we derive a new…
In this paper we generalize the involutive methods and algorithms devised for polynomial ideals to differential ones generated by a finite set of linear differential polynomials in the differential polynomial ring over a zero characteristic…
We consider a generalization of low-rank matrix completion to the case where the data belongs to an algebraic variety, i.e. each data point is a solution to a system of polynomial equations. In this case the original matrix is possibly…
We study the problem of bounding the posterior distribution of discrete probabilistic programs with unbounded support, loops, and conditioning. Loops pose the main difficulty in this setting: even if exact Bayesian inference is possible,…
In this work, we present a novel and practical approach to address one of the longstanding problems in computer vision: 2D and 3D affine invariant feature matching. Our Grassmannian Graph (GrassGraph) framework employs a two stage procedure…
General error locator polynomials are polynomials able to decode any correctable syndrome for a given linear code. Such polynomials are known to exist for all cyclic codes and for a large class of linear codes. We provide some decoding…
The affine inverse eigenvalue problem consists of identifying a real symmetric matrix with a prescribed set of eigenvalues in an affine space. Due to its ubiquity in applications, various instances of the problem have been widely studied in…
In this paper, we explore different techniques to synthesize modular invariants for synchronous code encoded as Horn clauses. Modular invariants are a set of formulas that characterizes the validity of predicates. They are very useful for…
The usual methods for root finding of polynomials are based on the iteration of a numerical formula for improvement of successive estimations. The unpredictable nature of the iterations prevents to search roots inside a pre-specified region…
We present a new method for inferring complexity properties for a class of programs in the form of flowcharts annotated with loop information. Specifically, our method can (soundly and completely) decide if computed values are polynomially…
We study homomorphism polynomials, which are polynomials that enumerate all homomorphisms from a pattern graph $H$ to $n$-vertex graphs. These polynomials have received a lot of attention recently for their crucial role in several new…
Verifiers that can prove programs correct against their full functional specification require, for programs with loops, additional annotations in the form of loop invariants---propeties that hold for every iteration of a loop. We show that…