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Formal verification of trading in financial marketsJul 18 2019We introduce a formal framework for analyzing trades in financial markets. An exchange is where multiple buyers and sellers participate to trade. These days, all big exchanges use computer algorithms that implement double sided auctions to match buy and ... More
A probabilistic approach to extreme statistics of Brownian escape times in dimensions 1, 2, and 3Jul 17 2019First passage time (FPT) theory is often used to estimate timescales in cellular and molecular biology. While the overwhelming majority of studies have focused on the time it takes a given single Brownian searcher to reach a target, cellular processes ... More
Logic Conditionals, Supervenience, and Selection TasksJul 15 2019Principles of cognitive economy would require that concepts about objects, properties and relations should be introduced only if they simplify the conceptualisation of a domain. Unexpectedly, classic logic conditionals, specifying structures holding within ... More
Topological rewriting systems applied to standard bases and syntactic algebrasJul 15 2019We propose a functional description of rewriting systems on topological vector spaces. We introduce the topological confluence property as an approximation of the confluence property. Using a representation of linear topological rewriting systems with ... More
The SAT+CAS Method for Combinatorial Search with Applications to Best MatricesJul 11 2019In this paper, we provide an overview of the SAT+CAS method that combines satisfiability checkers (SAT solvers) and computer algebra systems (CAS) to resolve combinatorial conjectures, and present new results vis-\`a-vis best matrices. The SAT+CAS method ... More
Computing the Maximum Degree of Minors in Skew Polynomial MatricesJul 10 2019Skew polynomials, which have a noncommutative multiplication rule between coefficients and an indeterminate, are the most general polynomial concept that admits the degree function with desirable properties. This paper presents the first algorithms to ... More
Improved Structural Methods for Nonlinear Differential-Algebraic Equations via Combinatorial RelaxationJul 10 2019Differential-algebraic equations (DAEs) are widely used for modeling of dynamical systems. In numerical analysis of DAEs, consistent initialization and index reduction are important preprocessing prior to numerical integration. Existing DAE solvers commonly ... More
SAT Solvers and Computer Algebra Systems: A Powerful Combination for MathematicsJul 09 2019Over the last few decades, many distinct lines of research aimed at automating mathematics have been developed, including computer algebra systems (CASs) for mathematical modelling, automated theorem provers for first-order logic, SAT/SMT solvers aimed ... More
Proving Properties of Sorting Programs: A Case Study in Horn Clause VerificationJul 09 2019The proof of a program property can be reduced to the proof of satisfiability of a set of constrained Horn clauses (CHCs) which can be automatically generated from the program and the property. In this paper we have conducted a case study in Horn clause ... More
Annotary: A Concolic Execution System for Developing Secure Smart ContractsJul 08 2019Ethereum smart contracts are executable programs, deployed on a peer-to-peer network and executed in a consensus-based fashion. Their bytecode is public, immutable and once deployed to the blockchain, cannot be patched anymore. As smart contracts may ... More
Solving p-adic polynomial systems via iterative eigenvector algorithmsJul 08 2019In this article, we describe an implementation of a polynomial system solver to compute the approximate solutions of a 0-dimensional polynomial system with finite precision p-adic arithmetic. We also describe an improvement to an algorithm of Caruso, ... More
Automatic Differentiation for Adjoint Stencil LoopsJul 05 2019Stencil loops are a common motif in computations including convolutional neural networks, structured-mesh solvers for partial differential equations, and image processing. Stencil loops are easy to parallelise, and their fast execution is aided by compilers, ... More
Absolute root separationJul 02 2019The absolute separation of a polynomial is the minimum nonzero difference between the absolute values of its roots. In the case of polynomials with integer coefficients, it can be bounded from below in terms of the degree and the height (the maximum absolute ... More
Solving Polynomial Systems with phcpyJun 28 2019The solutions of a system of polynomials in several variables are often needed, e.g.: in the design of mechanical systems, and in phase-space analyses of nonlinear biological dynamics. Reliable, accurate, and comprehensive numerical solutions are available ... More
Semantic Preserving Bijective Mappings for Expressions involving Special Functions in Computer Algebra Systems and Document Preparation SystemsJun 27 2019Purpose: Modern mathematicians and scientists of math-related disciplines often use Document Preparation Systems (DPS) to write and Computer Algebra Systems (CAS) to calculate mathematical expressions. Usually, they translate the expressions manually ... More
Neurally-Guided Structure InferenceJun 17 2019Most structure inference methods either rely on exhaustive search or are purely data-driven. Exhaustive search robustly infers the structure of arbitrarily complex data, but it is slow. Data-driven methods allow efficient inference, but do not generalize ... More
Effective problem solving using SAT solversJun 14 2019In this article we demonstrate how to solve a variety of problems and puzzles using the built-in SAT solver of the computer algebra system Maple. Once the problems have been encoded into Boolean logic, solutions can be found (or shown to not exist) automatically, ... More
Neural Variational Inference For Estimating Uncertainty in Knowledge Graph EmbeddingsJun 12 2019Recent advances in Neural Variational Inference allowed for a renaissance in latent variable models in a variety of domains involving high-dimensional data. While traditional variational methods derive an analytical approximation for the intractable distribution ... More
Efficient Graph RewritingJun 11 2019Graph transformation is the rule-based modification of graphs, and is a discipline dating back to the 1970s. The declarative nature of graph rewriting rules comes at a cost. In general, to match the left-hand graph of a fixed rule within a host graph ... More
Polynomial root clustering and explicit deflationJun 11 2019We seek complex roots of a univariate polynomial $P$ with real or complex coefficients. We address this problem based on recent algorithms that use subdivision and have a nearly optimal complexity. They are particularly efficient when only roots in a ... More
New Features in the Second Version of the Cadabra Computer Algebra SystemJun 06 2019In certain scientific domains, there is a need for tensor operations. To facilitate tensor computations,computer algebra systems are employed. In our research, we have been using Cadabra as the main computer algebra system for several years. Recently, ... More
Standard Lattices of Compatibly Embedded Finite FieldsJun 03 2019Lattices of compatibly embedded finite fields are useful in computer algebra systems for managing many extensions of a finite field $\mathbb{F}_p$ at once. They can also be used to represent the algebraic closure $\bar{\mathbb{F}}_p$, and to represent ... More
Algorithmically generating new algebraic features of polynomial systems for machine learningJun 03 2019There are a variety of choices to be made in both computer algebra systems (CASs) and satisfiability modulo theory (SMT) solvers which can impact performance without affecting mathematical correctness. Such choices are candidates for machine learning ... More
Abstract Predicate Entailment over Points-To Heaplets is Syntax RecognitionJun 01 2019Abstract predicates are considered in this paper as abstraction technique for heap-separated configurations, and as genuine Prolog predicates which are translated straight into a corresponding formal language grammar used as validation scheme for intermediate ... More
On the Parallelization of Triangular Decomposition of Polynomial SystemsMay 31 2019We discuss the parallelization of algorithms for solving polynomial systems symbolically by way of triangular decomposition. Algorithms for solving polynomial systems combine low-level routines for performing arithmetic operations on polynomials and high-level ... More
An Algorithm for Computing Invariant Projectors in Representations of Wreath ProductsMay 29 2019We describe an algorithm for computing the complete set of primitive orthogonal idempotents in the centralizer ring of the permutation representation of a wreath product. This set of idempotents determines the decomposition of the representation into ... More
Factorizations for a Class of Multivariate Polynomial MatricesMay 28 2019Following the works by Lin et al. (Circuits Syst. Signal Process. 20(6): 601-618, 2001) and Liu et al. (Circuits Syst. Signal Process. 30(3): 553-566, 2011), we investigate how to factorize a class of multivariate polynomial matrices. The main theorem ... More
Reconstruction of rational ruled surfaces from their silhouettesMay 28 2019We provide algorithms to reconstruct rational ruled surfaces in three-dimensional projective space from the `apparent contour' of a single projection to the projective plane. We deal with the case of tangent developables and of general projections to ... More
Confluence by Critical Pair Analysis Revisited (Extended Version)May 28 2019Jun 03 2019We present two methods for proving confluence of left-linear term rewrite systems. One is hot-decreasingness, combining the parallel/development closedness theorems with rule labelling based on a terminating subsystem. The other is critical-pair-closing ... More
Confluence by Critical Pair Analysis Revisited (Extended Version)May 28 2019We present two methods for proving confluence of left-linear term rewrite systems. One is hot-decreasingness, combining the parallel/development closedness theorems with rule labelling based on a terminating subsystem. The other is critical-pair-closing ... More
A closed-form formula for the Kullback-Leibler divergence between Cauchy distributionsMay 27 2019May 28 2019We report a closed-form expression for the Kullback-Leibler divergence between Cauchy distributions which involves the calculation of a novel definite integral. The formula shows that the Kullback-Leibler divergence between Cauchy densities is always ... More
New ways to multiply 3 x 3-matricesMay 24 2019It is known since the 1970s that no more than 23 multiplications are required for computing the product of two 3 x 3-matrices. It is not known whether this can also be done with fewer multiplications. However, there are several mutually inequivalent ways ... More
Lonely Points in SimplicesMay 21 2019Given a lattice L in Z^m and a subset A of R^m, we say that a point in A is lonely if it is not equivalent modulo L to another point of A. We are interested in identifying lonely points for specific choices of L when A is a dilated standard simplex, and ... More
A polynomial approach to the Collatz conjectureMay 21 2019The Collatz conjecture is explored using polynomials based on a binary numeral system. It is shown that the degree of the polynomials, on average, decreases after a finite number of steps of the Collatz operation, which provides a weak proof of the conjecture ... More
Computing symmetric determinantal representationsMay 16 2019We introduce the DeterminantalRepresentations package for Macaulay2, which computes definite symmetric determinantal representations of real polynomials. We focus on quadrics and plane curves of low degree (i.e. cubics and quartics). Our algorithms are ... More
A Correctness Result for Synthesizing Plans With Loops in Stochastic DomainsMay 16 2019Finite-state controllers (FSCs), such as plans with loops, are powerful and compact representations of action selection widely used in robotics, video games and logistics. There has been steady progress on synthesizing FSCs in deterministic environments, ... More
Fairness in Machine Learning with Tractable ModelsMay 16 2019Machine Learning techniques have become pervasive across a range of different applications, and are now widely used in areas as disparate as recidivism prediction, consumer credit-risk analysis and insurance pricing. The prevalence of machine learning ... More
Effects Without Monads: Non-determinism -- Back to the Meta LanguageMay 16 2019We reflect on programming with complicated effects, recalling an undeservingly forgotten alternative to monadic programming and checking to see how well it can actually work in modern functional languages. We adopt and argue the position of factoring ... More
Critical points at infinity for analytic combinatoricsMay 13 2019On complex algebraic varieties, height functions arising in combinatorial applications fail to be proper. This complicates the description and computation via Morse theory of key topological invariants. Here we establish checkable conditions under which ... More
Change of basis for m-primary ideals in one and two variablesMay 12 2019Following recent work by van der Hoeven and Lecerf (ISSAC 2017), we discuss the complexity of linear mappings, called untangling and tangling by those authors, that arise in the context of computations with univariate polynomials. We give a slightly faster ... More
Implementations of efficient univariate polynomial matrix algorithms and application to bivariate resultantsMay 10 2019Complexity bounds for many problems on matrices with univariate polynomial entries have been improved in the last few years. Still, for most related algorithms, efficient implementations are not available, which leaves open the question of the practical ... More
Integrality and arithmeticity of solvable linear groupsMay 10 2019We develop a practical algorithm to decide whether a finitely generated subgroup of a solvable algebraic group $G$ is arithmetic. This incorporates a procedure to compute a generating set of an arithmetic subgroup of $G$. We also provide a simple new ... More
Effective Coefficient Asymptotics of Multivariate Rational Functions via Semi-Numerical Algorithms for Polynomial SystemsMay 10 2019The coefficient sequences of multivariate rational functions appear in many areas of combinatorics. Their diagonal coefficient sequences enjoy nice arithmetic and asymptotic properties, and the field of analytic combinatorics in several variables (ACSV) ... More
Asymptotics of multivariate sequences in the presence of a lacunaMay 10 2019We explain a discontinuous drop in the exponential growth rate for certain multivariate generating functions at a critical parameter value, in even dimensions $d \geq 4$. This result depends on computations in the homology of the algebraic variety where ... More
An Algorithmic Approach to Limit Cycles of Nonlinear Differential Systems: the Averaging Method RevisitedMay 08 2019This paper introduces an algorithmic approach to the analysis of bifurcation of limit cycles from the centers of nonlinear continuous differential systems via the averaging method. We develop three algorithms to implement the averaging method. The first ... More
Automatic Generation of Moment-Based Invariants for Prob-Solvable LoopsMay 07 2019May 29 2019One of the main challenges in the analysis of probabilistic programs is to compute invariant properties that summarise loop behaviours. Automation of invariant generation is still at its infancy and most of the times targets only expected values of the ... More
Automatic Generation of Moment-Based Invariants for Prob-Solvable LoopsMay 07 2019May 21 2019One of the main challenges in the analysis of probabilistic programs is to compute invariant properties that summarise loop behaviours. Automation of invariant generation is still at its infancy and most of the times targets only expected values of the ... More
Automatic Generation of Moment-Based Invariants for Prob-Solvable LoopsMay 07 2019One of the main challenges in the analysis of probabilistic programs is to compute invariant properties that summarise loop behaviours. Automation of invariant generation is still at its infancy and most of the times targets only expected values of the ... More
The strong approximation theorem and computing with linear groupsMay 07 2019We obtain a computational realization of the strong approximation theorem. That is, we develop algorithms to compute all congruence quotients modulo rational primes of a finitely generated Zariski dense group $H \leq \mathrm{SL}(n, \mathbb{Z})$ for $n ... More
The complexity of MinRankMay 06 2019Jun 03 2019In this note, we leverage some of our results from arXiv:1706.06319 to produce a concise and rigorous proof for the complexity of the generalized MinRank Problem in the under-defined and well-defined case. Our main theorem recovers and extends previous ... More
The complexity of MinRankMay 06 2019In this note, we leverage some of our results from arXiv:1706.06319 to produce a concise and rigorous proof for the complexity of the generalized MinRank Problem in the under-defined and well-defined case. Our main theorem recovers and extends previous ... More
Derandomization from Algebraic Hardness: Treading the BordersApr 30 2019A hitting-set generator (HSG) is a polynomial map $Gen:\mathbb{F}^k \to \mathbb{F}^n$ such that for all $n$-variate polynomials $Q$ of small enough circuit size and degree, if $Q$ is non-zero, then $Q\circ Gen$ is non-zero. In this paper, we give a new ... More
Algorithmic approach to strong consistency analysis of finite difference approximations to PDE systemsApr 29 2019For a wide class of polynomially nonlinear systems of partial differential equations we suggest an algorithmic approach to the s(trong)-consistency analysis of their finite difference approximations on Cartesian grids. First we apply the differential ... More
Computing the volume of compact semi-algebraic setsApr 26 2019Let $S\subset R^n$ be a compact basic semi-algebraic set defined as the real solution set of multivariate polynomial inequalities with rational coefficients. We design an algorithm which takes as input a polynomial system defining $S$ and an integer $p\geq ... More
Constructing minimal telescopers for rational functions in three discrete variablesApr 25 2019We present a new algorithm for constructing minimal telescopers for rational functions in three discrete variables. This is the first discrete reduction-based algorithm that goes beyond the bivariate case. The termination of the algorithm is guaranteed ... More
Constructing minimal telescopers for rational functions in three discrete variablesApr 25 2019May 03 2019We present a new algorithm for constructing minimal telescopers for rational functions in three discrete variables. This is the first discrete reduction-based algorithm that goes beyond the bivariate case. The termination of the algorithm is guaranteed ... More
Comparing machine learning models to choose the variable ordering for cylindrical algebraic decompositionApr 24 2019There has been recent interest in the use of machine learning (ML) approaches within mathematical software to make choices that impact on the computing performance without affecting the mathematical correctness of the result. We address the problem of ... More
Comparing machine learning models to choose the variable ordering for cylindrical algebraic decompositionApr 24 2019Jun 05 2019There has been recent interest in the use of machine learning (ML) approaches within mathematical software to make choices that impact on the computing performance without affecting the mathematical correctness of the result. We address the problem of ... More
$\mathtt{bimEX}$: A Mathematica package for exact computations in 3$+$1 bimetric relativityApr 23 2019We present $\mathtt{bimEX}$, a Mathematica package for exact computations in 3$+$1 bimetric relativity. It is based on the $\mathtt{xAct}$ bundle, which can handle computations involving both abstract tensors and their components. In this communication, ... More
Improved algorithms for left factorial residuesApr 19 2019We present improved algorithms for computing the left factorial residues $!p=0!+1!+...+(p-1)! \!\mod p$. We use these algorithms for the calculation of the residues $!p\!\mod p$, for all primes $p$ up to $2^{40}$. Our results confirm that Kurepa's left ... More
Unification and combination of iterative insertion strategies with one-step traversalsApr 16 2019Motivated by an ongoing project on the computer aided derivation of multiscale partial differential equation models, we introduce a class of term transformations that consists in navigation strategies and insertion of contexts. We define a unification ... More
Unification and combination of iterative insertion strategies with rudimentary traversals and failureApr 16 2019We introduce a new class of extensions of terms that consists in navigation strategies and insertion of contexts. We introduce an operation of combination on this class which is associative, admits a neutral element and so that each extension is idempotent. ... More
Asymptotic Solutions of Polynomial Equations with Exp-Log CoefficientsApr 15 2019We present an algorithm for computing asymptotic approximations of roots of polynomials with exp-log function coefficients. The real and imaginary parts of the approximations are given as explicit exp-log expressions. We provide a method for deciding ... More
Proceedings Joint International Workshop on Linearity & Trends in Linear Logic and ApplicationsApr 12 2019This volume contains a selection of papers presented at Linearity/TLLA 2018: Joint Linearity and TLLA workshops (part of FLOC 2018) held on July 7-8, 2018 in Oxford. Linearity has been a key feature in several lines of research in both theoretical and ... More
On the Equivalence of Forward Mode Automatic Differentiation and Symbolic DifferentiationApr 05 2019May 03 2019We show that forward mode automatic differentiation and symbolic differentiation are equivalent in the sense that they both perform the same operations when computing derivatives. This is in stark contrast to the common claim that they are substantially ... More
Tropical Differential Groebner BasisApr 03 2019In this paper, the tropical differential Gr\"obner basis is studied, which is a natural generalization of the tropical Gr\"obner basis to the recently introduced tropical differential algebra. Like the differential Gr\"obner basis, the tropical differential ... More
Exact Lower Bounds for Monochromatic Schur Triples and GeneralizationsApr 03 2019We derive exact and sharp lower bounds for the number of monochromatic generalized Schur triples $(x,y,x+ay)$ whose entries are from the set $\{1,\dots,n\}$, subject to a coloring with two different colors. Previously, only asymptotic formulas for such ... More
Reconstructing Rational Functions with $\texttt{FireFly}$Mar 29 2019We present the open-source $\texttt{C++}$ library $\texttt{FireFly}$ for the reconstruction of multivariate rational functions over finite fields. We discuss the involved algorithms and their implementation. As an application, we use $\texttt{FireFly}$ ... More
Computing huge Groebner basis like cyclic10 over $\Q$ with GiacMar 29 2019We present a short description on how to fine-tune the modular algorithm implemented in the Giac computer algebra system to reconstruct huge Groebner basis over $\Q$.The classical cyclic10 benchmark will serve as example.
Testing zero-dimensionality of varieties at a pointMar 29 2019Effective methods are introduced for testing zero-dimensionality of varieties at a point. The motivation of this paper is to compute and analyze deformations of isolated hypersurface singularities. As an application, methods for computing local dimensions ... More
Local Search for Fast Matrix MultiplicationMar 27 2019Laderman discovered a scheme for computing the product of two 3x3 matrices using only 23 multiplications in 1976. Since then, some more such schemes were proposed, but it remains open how many there are and whether there exist schemes with fewer than ... More
Computations of the Expected Euler Characteristic for the Largest Eigenvalue of a Real Wishart MatrixMar 25 2019We give an approximate formula of the distribution of the largest eigenvalue of real Wishart matrices by the expected Euler characteristic method for the general dimension. The formula is expressed in terms of a definite integral with parameters. We derive ... More
Cylindrical Algebraic Decomposition with Equational ConstraintsMar 20 2019Cylindrical Algebraic Decomposition (CAD) has long been one of the most important algorithms within Symbolic Computation, as a tool to perform quantifier elimination in first order logic over the reals. More recently it is finding prominence in the Satisfiability ... More
On some classes of irreducible polynomialsMar 20 2019The aim of the paper is to produce new families of irreducible polynomials, generalizing previous results in the area. One example of our general result is that for a near-separated polynomial, i.e., polynomials of the form $F(x,y)=f_1(x)f_2(y)-f_2(x)f_1(y)$, ... More
Minimizing polynomial functions on quantum computersMar 19 2019This expository paper reviews some of the recent uses of computational algebraic geometry in classical and quantum optimization. The paper assumes an elementary background in algebraic geometry and adiabatic quantum computing (AQC), and concentrates on ... More
Recursive Matrix Algorithms in Commutative Domain for Cluster with Distributed MemoryMar 11 2019We give an overview of the theoretical results for matrix block-recursive algorithms in commutative domains and present the results of experiments that we conducted with new parallel programs based on these algorithms on a supercomputer MVS-10P at the ... More
Quadratic Probabilistic Algorithms for Normal BasesMar 08 2019It is well known that for any finite Galois extension field $K/F$, with Galois group $G = \mathrm{Gal}(K/F)$, there exists an element $\alpha \in K$ whose orbit $G\cdot\alpha$ forms an $F$-basis of $K$. Such an element $\alpha$ is called \emph{normal} ... More
Discovering and Proving Infinite Pochhammer Sum IdentitiesFeb 28 2019We consider nested sums involving the Pochhammer symbol at infinity and rewrite them in terms of a small set of constants, such as powers of $\pi,$ $\log(2)$ or zeta values. In order to perform these simplifications, we view the series as specializations ... More
Discovering and Proving Infinite Pochhammer Sum IdentitiesFeb 28 2019Apr 10 2019We consider nested sums involving the Pochhammer symbol at infinity and rewrite them in terms of a small set of constants, such as powers of $\pi,$ $\log(2)$ or zeta values. In order to perform these simplifications, we view the series as specializations ... More
Counting basic-irreducible factors mod $p^k$ in deterministic poly-time and $p$-adic applicationsFeb 20 2019Finding an irreducible factor, of a polynomial $f(x)$ modulo a prime $p$, is not known to be in deterministic polynomial time. Though there is such a classical algorithm that {\em counts} the number of irreducible factors of $f\bmod p$. We can ask the ... More
Computing Minimal Presentations and Betti Numbers of 2-Parameter Persistent HomologyFeb 15 2019Motivated by applications to topological data analysis, we give an efficient algorithm for computing a (minimal) presentation of a bigraded $K[x,y]$-module $M$, where $K$ is a field. The algorithm assumes that $M$ is given implicitly: It takes as input ... More
Computing Minimal Presentations and Bigraded Betti Numbers of 2-Parameter Persistent HomologyFeb 15 2019Mar 25 2019Motivated by applications to topological data analysis, we give an efficient algorithm for computing a (minimal) presentation of a bigraded $K[x,y]$-module $M$, where $K$ is a field. The algorithm takes as input a short chain complex of free modules \[ ... More
Identifying the Parametric Occurrence of Multiple Steady States for some Biological NetworksFeb 13 2019We consider a problem from biological network analysis of determining regions in a parameter space over which there are multiple steady states for positive real values of variables and parameters. We describe multiple approaches to address the problem ... More
When Causal Intervention Meets Image Masking and Adversarial Perturbation for Deep Neural NetworksFeb 09 2019Feb 13 2019Discovering and exploiting the causality in deep neural networks (DNNs) are crucial challenges for understanding and reasoning causal effects (CE) on an explainable visual model. "Intervention" has been widely used for recognizing a causal relation ontologically. ... More
When Causal Intervention Meets Image Masking and Adversarial Perturbation for Deep Neural NetworksFeb 09 2019Discovering and exploiting the causality in deep neural networks (DNNs) are crucial challenges for understanding and reasoning causal effects (CE) on an explainable visual model. "Intervention" has been widely used for recognizing a causal relation ontologically. ... More
When Causal Intervention Meets Adversarial Examples and Image Masking for Deep Neural NetworksFeb 09 2019Jun 25 2019Discovering and exploiting the causality in deep neural networks (DNNs) are crucial challenges for understanding and reasoning causal effects (CE) on an explainable visual model. "Intervention" has been widely used for recognizing a causal relation ontologically. ... More
Generic reductions for in-place polynomial multiplicationFeb 08 2019The polynomial multiplication problem has attracted considerable attention since the early days of computer algebra, and several algorithms have been designed to achieve the best possible time complexity. More recently, efforts have been made to improve ... More
Exact Optimization via Sums of Nonnegative Circuits and Sums of AM/GM ExponentialsFeb 06 2019We provide two hybrid numeric-symbolic optimization algorithms, computing exact sums of nonnegative circuits (SONC) and sums of arithmetic-geometric-exponentials (SAGE) decompositions. Moreover, we provide a hybrid numeric-symbolic decision algorithm ... More
Modeling Terms by Graphs with Structure Constraints (Two Illustrations)Feb 06 2019In the talk at the workshop my aim was to demonstrate the usefulness of graph techniques for tackling problems that have been studied predominantly as problems on the term level: increasing sharing in functional programs, and addressing questions about ... More
On the Complexity of Toric IdealsFeb 04 2019We investigate the computational complexity of problems on toric ideals such as normal forms, Gr\"obner bases, and Graver bases. We show that all these problems are strongly NP-hard in the general case. Nonetheless, we can derive efficient algorithms ... More
Gr{ö}bner Basis over Semigroup Algebras: Algorithms and Applications for Sparse Polynomial SystemsFeb 01 2019Gr{\"o}bner bases is one the most powerful tools in algorithmic non-linear algebra. Their computation is an intrinsically hard problem with a complexity at least single exponential in the number of variables. However, in most of the cases, the polynomial ... More
Algorithmic counting of nonequivalent compact Huffman codesJan 31 2019It is known that the following five counting problems lead to the same integer sequence~$f_t(n)$: the number of nonequivalent compact Huffman codes of length~$n$ over an alphabet of $t$ letters, the number of `nonequivalent' canonical rooted $t$-ary trees ... More
Rational Solutions of First-Order Algebraic Ordinary Difference EquationsJan 30 2019Feb 01 2019We propose an algebraic geometric approach for studying rational solutions of first-order algebraic ordinary difference equations. For an autonomous first-order algebraic ordinary difference equations, we give an upper bound for the degrees of its rational ... More
LU factorization with errors *Jan 30 2019We present new algorithms to detect and correct errors in the lower-upper factorization of a matrix, or the triangular linear system solution, over an arbitrary field. Our main algorithms do not require any additional information or encoding other than ... More
An Optimization-Based Sum-of-Squares Approach to Vizing's ConjectureJan 29 2019Feb 04 2019Vizing's conjecture (open since 1968) relates the sizes of dominating sets in two graphs to the size of a dominating set in their Cartesian product graph. In this paper, we formulate Vizing's conjecture itself as a Positivstellensatz existence question. ... More
An Optimization-Based Sum-of-Squares Approach to Vizing's ConjectureJan 29 2019May 06 2019Vizing's conjecture (open since 1968) relates the sizes of dominating sets in two graphs to the size of a dominating set in their Cartesian product graph. In this paper, we formulate Vizing's conjecture itself as a Positivstellensatz existence question. ... More
A Faster Solution to Smale's 17th Problem I: Real Binomial SystemsJan 28 2019Suppose $F:=(f_1,\ldots,f_n)$ is a system of random $n$-variate polynomials with $f_i$ having degree $\leq\!d_i$ and the coefficient of $x^{a_1}_1\cdots x^{a_n}_n$ in $f_i$ being an independent complex Gaussian of mean $0$ and variance $\frac{d_i!}{a_1!\cdots ... More
On the Complexity of Computing the Topology of Real Algebraic Space CurvesJan 28 2019In this paper, we present a deterministic algorithm to find a strong generic position for an algebraic space curve. We modify our existing algorithm for computing the topology of an algebraic space curve and analyze the bit complexity of the algorithm. ... More
Signature-based Möller's algorithm for strong Gröbner bases over PIDsJan 28 2019Signature-based algorithms are the latest and most efficient approach as of today to compute Gr\"obner bases for polynomial systems over fields. Recently, possible extensions of these techniques to general rings have attracted the attention of several ... More
Gr{ö}bner bases over Tate algebrasJan 28 2019Tate algebras are fundamental objects in the context of analytic geometry over the p-adics. Roughly speaking, they play the same role as polynomial algebras play in classical algebraic geometry. In the present article, we develop the formalism of Gr{\"o}bner ... More