Computing the Newton-step of a generic function with $N$ decision variables takes $O(N^3)$ flops. In this paper, we show that given the computational graph of the function, this bound can be reduced to $O(m\tau^3)$, where $\tau, m$ are the width and size of a tree-decomposition of the graph. The proposed algorithm generalizes nonlinear optimal-control methods based on LQR to general optimization problems

Signature-based algorithms have become a standard approach for computing Gr\"obner bases in commutative polynomial rings. However, so far, it was not clear how to extend this concept to the setting of noncommutative polynomials in the free algebra. In this paper, we present a signature-based algorithm for computing Gr\"obner bases in precisely this setting. The algorithm is an adaptation of Buchberger's

While there are numerous linear algebra teaching tools, they tend to be focused on the basics, and not handle the more advanced aspects. This project aims to fill that gap, focusing specifically on methods like Strassen's fast matrix multiplication.

Some aspects of Computer Algebra (notably Computation Group Theory and Computational Number Theory) have some good databases of examples, typically of the form "all the X up to size n". But most of the others, especially on the polynomial side, are lacking such, despite the utility they have demonstrated in the related fields of SAT and SMT solving. We claim that the field would be enhanced by such

Assessing non-negativity of multivariate polynomials over the reals, through the computation of {\em certificates of non-negativity}, is a topical issue in polynomial optimization. This is usually tackled through the computation of {\em sums-of-squares decompositions} which rely on efficient numerical solvers for semi-definite programming. This method faces two difficulties. The first one is that the

Compression aims to reduce the size of an input, while maintaining its relevant properties. For multi-parameter persistent homology, compression is a necessary step in any computational pipeline, since standard constructions lead to large inputs, and computational tasks in this area tend to be expensive. We propose two compression methods for chain complexes of free 2-parameter persistence modules

Saturation-style automated theorem provers (ATPs) based on the given clause procedure are today the strongest general reasoners for classical first-order logic. The clause selection heuristics in such systems are, however, often evaluating clauses in isolation, ignoring other clauses. This has changed recently by equipping the E/ENIGMA system with a graph neural network (GNN) that chooses the next

We give partial generalizations of the classical Descartes' rule of signs to multivariate polynomials (with real exponents), in the sense that we provide upper bounds on the number of connected components of the complement of a hypersurface in the positive orthant. In particular, we give conditions based on the geometrical configuration of the exponents and the sign of the coefficients that guarantee

This paper introduces a differentiable semantic reasoner, where rules are presented as a relevant set of graph transformations. These rules can be written manually or inferred by a set of facts and goals presented as a training set. While the internal representation uses embeddings in a latent space, each rule can be expressed as a set of predicates conforming to a subset of Description Logic.

Physics-informed neural networks (PINNs) are an increasingly powerful way to solve partial differential equations, generate digital twins, and create neural surrogates of physical models. In this manuscript we detail the inner workings of NeuralPDE.jl and show how a formulation structured around numerical quadrature gives rise to new loss functions which allow for adaptivity towards bounded error tolerances

In research problems that involve the use of numerical methods for solving systems of ordinary differential equations (ODEs), it is often required to select the most efficient method for a particular problem. To solve a Cauchy problem for a system of ODEs, Runge-Kutta methods (explicit or implicit ones, with or without step-size control, etc.) are employed. In that case, it is required to search through

Despite the groundbreaking successes of neural networks, contemporary models require extensive training with massive datasets and exhibit poor out-of-sample generalization. One proposed solution is to build systematicity and domain-specific constraints into the model, echoing the tenets of classical, symbolic cognitive architectures. In this paper, we consider the limitations of this approach by examining

Sparse polynomial interpolation, sparse linear system solving or modular rational reconstruction are fundamental problems in Computer Algebra. They come down to computing linear recurrence relations of a sequence with the Berlekamp-Massey algorithm. Likewise, sparse multivariate polynomial interpolation and multidimensional cyclic code decoding require guessing linear recurrence relations of a multivariate

We consider the problem of binomiality of the steady state ideals of biochemical reaction networks. We are interested in finding polynomial conditions on the parameters such that the steady state ideal of a chemical reaction network is binomial under every specialisation of the parameters if the conditions on the parameters hold. We approach the binomiality problem using Comprehensive Gr\"obner systems

Parameter identifiability describes whether, for a given differential model, one can determine parameter values from model equations. Knowing global or local identifiability properties allows construction of better practical experiments to identify parameters from experimental data. In this work, we present a web-based software tool that allows to answer specific identifiability queries. Concretely

We present MultivariatePowerSeries, a Maple library introduced in Maple 2021, providing a variety of methods to study formal multivariate power series and univariate polynomials over such series. This library offers a simple and easy-to-use user interface. Its implementation relies on lazy evaluation techniques and takes advantage of Maple's features for object-oriented programming. The exposed methods

We design a fast algorithm that computes, for a given linear differential operator with coefficients in Z[ ], all the characteristic polynomials of its-curvatures, for all primes < , in asymptotically quasi-linear bit complexity in. We discuss implementations and applications of our algorithm. We shall see in particular that the good performances of our algorithm are quickly visible. CCS CONCEPTS $\bullet$

We propose an algorithm to compute the $C^\infty$-ring structure of arbitrary Weil algebra. It allows us to do some analysis with higher infinitesimals numerically and symbolically. To that end, we first give a brief description of the (Forward-mode) automatic differentiation (AD) in terms of $C^\infty$-rings. The notion of a $C^\infty$-ring was introduced by Lawvere and used as the fundamental building

We describe a new algorithm for computing Whitney stratifications of complex projective varieties. The main ingredients are (a) an algebraic criterion, due to L\^e and Teissier, which reformulates Whitney regularity in terms of conormal spaces and maps, and (b) a new interpretation of this conormal criterion via ideal saturations, which can be practically implemented on a computer. We show that this

Symbolic regression is the task of identifying a mathematical expression that best fits a provided dataset of input and output values. Due to the richness of the space of mathematical expressions, symbolic regression is generally a challenging problem. While conventional approaches based on genetic evolution algorithms have been used for decades, deep learning-based methods are relatively new and an

We study the problem of recovering a collection of $n$ numbers from the evaluation of $m$ power sums. This yields a system of polynomial equations, which can be underconstrained ($m < n$), square ($m = n$), or overconstrained ($m > n$). Fibers and images of power sum maps are explored in all three regimes, and in settings that range from complex and projective to real and positive. This involves surprising

This article does not introduce new mathematical ideas, it is more a state of the art June 2021 picture about solving polynomial systems efficiently by reconstructing a rational univariate representation with a very high probability of correctness using Groebner revlex computation, Berlekamp-Massey algorithm and Hankel linear system solving modulo several primes in parallel. This algorithm is implemented

We improve the current best running time value to invert sparse matrices over finite fields, lowering it to an expected $O\big(n^{2.2131}\big)$ time for the current values of fast rectangular matrix multiplication. We achieve the same running time for the computation of the rank and nullspace of a sparse matrix over a finite field. This improvement relies on two key techniques. First, we adopt the

Entity linking (EL), the task of disambiguating mentions in text by linking them to entities in a knowledge graph, is crucial for text understanding, question answering or conversational systems. Entity linking on short text (e.g., single sentence or question) poses particular challenges due to limited context. While prior approaches use either heuristics or black-box neural methods, here we propose

We design algorithms for computing values of many p-adic elementary and special functions, including logarithms, exponentials, polylogarithms, and hypergeometric functions. All our algorithms feature a quasi-linear complexity with respect to the target precision and most of them are based on an adaptation to the-adic setting of the binary splitting and bit-burst strategies.

This abstract seeks to introduce the ISSAC community to the DEWCAD project, which is based at Coventry University and the University of Bath, in the United Kingdom. The project seeks to push back the Doubly Exponential Wall of Cylindrical Algebraic Decomposition, through the integration of SAT/SMT technology, the extension of Lazard projection theory, and the development of new algorithms based on

Artificial Neural Networks (ANNs) are being deployed on an increasing number of safety-critical applications, including autonomous cars and medical diagnosis. However, concerns about their reliability have been raised due to their black-box nature and apparent fragility to adversarial attacks. Here, we develop and evaluate a symbolic verification framework using incremental model checking (IMC) and

We present a new model-based interpolation procedure for satisfiability modulo theories (SMT). The procedure uses a new mode of interaction with the SMT solver that we call solving modulo a model. This either extends a given partial model into a full model for a set of assertions or returns an explanation (a model interpolant) when no solution exists. This mode of interaction fits well into the model-constructing

We present a number of new piecewise-polynomial kernels for image interpolation. The kernels are constructed by optimizing a measure of interpolation quality based on the magnitude of anisotropic artifacts. The kernel design process is performed symbolically using Mathematica computer algebra system. Experimental evaluation involving 14 image quality assessment methods demonstrates that our results

What can be (machine) learned about the complexity of Buchberger's algorithm? Given a system of polynomials, Buchberger's algorithm computes a Gr\"obner basis of the ideal these polynomials generate using an iterative procedure based on multivariate long division. The runtime of each step of the algorithm is typically dominated by a series of polynomial additions, and the total number of these additions

We present a new data structure to approximate accurately and efficiently a polynomial $f$ of degree $d$ given as a list of coefficients. Its properties allow us to improve the state-of-the-art bounds on the bit complexity for the problems of root isolation and approximate multipoint evaluation. This data structure also leads to a new geometric criterion to detect ill-conditioned polynomials, implying

The article proposes a computer program for calculating economic crises according to the generalized mathematical model of S.V. Dubovsky. This model is represented by a system of ordinary nonlinear differential equations with fractional derivatives in the sense of Gerasimov-Caputo with initial conditions. Furthermore, according to a numerical algorithm based on an explicit nonlocal finite-difference

We consider resultant-based methods for elimination of indeterminates of Ore polynomial systems in Ore algebra. We start with defining the concept of resultant for bivariate Ore polynomials then compute it by the Dieudonne determinant of the polynomial coefficients. Additionally, we apply noncommutative versions of evaluation and interpolation techniques to the computation process to improve the efficiency

Effective computation of resultants is a central problem in elimination theory and polynomial system solving. Commonly, we compute the resultant as a quotient of determinants of matrices and we say that there exists a determinantal formula when we can express it as a determinant of a matrix whose elements are the coefficients of the input polynomials. We study the resultant in the context of mixed

Arabshahi, Singh, and Anandkumar (2018) propose a method for creating a dataset of symbolic mathematical equations for the tasks of symbolic equation verification and equation completion. Unfortunately, a dataset constructed using the method they propose will suffer from two serious flaws. First, the class of true equations that the procedure can generate will be very limited. Second, because true

Hensel's lemma, combined with repeated applications of Weierstrass preparation theorem, allows for the factorization of polynomials with multivariate power series coefficients. We present a complexity analysis for this method and leverage those results to guide the load-balancing of a parallel implementation to concurrently update all factors. In particular, the factorization creates a pipeline where

We study real steady state varieties of the dynamics of chemical reaction networks. The dynamics are derived using mass action kinetics with parametric reaction rates. The models studied are not inherently parametric in nature. Rather, our interest in parameters is motivated by parameter uncertainty, as reaction rates are typically either measures with limited precision or estimated. We aim at detecting

Let $V$ be the set of real common solutions to $F = (f_1, \ldots, f_s)$ in $\mathbb{R}[x_1, \ldots, x_n]$ and $D$ be the maximum total degree of the $f_i$'s. % The We design an algorithm which on input $F$ computes the dimension of $V$. Letting $L$ be the evaluation complexity of $F$ and $s=1$, it runs using $O^\sim \big (L D^{n(d+3)+1}\big )$ arithmetic operations in $\mathbb{Q}$ and at most $D^{n(d+1)}$

From the literature it is known that orthogonal polynomials as the Jacobi polynomials can be expressed by hypergeometric series. In this paper, the authors derive several contiguous relations for terminating multivariate hypergeometric series. With these contiguous relations one can prove several recursion formulas of those series. This theoretical result allows to compute integrals over products of

Two operads are said to belong to the same Wilf class if they have the same generating series. We discuss possible Wilf classifications of non-symmetric operads with monomial relations. As a corollary, this would give the same classification for the operads with a finite Groebner basis. Generally, there is no algorithm to decide whether two finitely presented operads belong to the same Wilf class.

We present and prove closed form expressions for some families of binomial determinants with signed Kronecker deltas that are located along an arbitrary diagonal in the corresponding matrix. They count cyclically symmetric rhombus tilings of hexagonal regions with triangular holes. We extend a previous systematic study of these families, where the locations of the Kronecker deltas depended on an additional

Solving nonlinear systems is an important problem. Numerical continuation methods efficiently solve certain nonlinear systems. The Asymptotic Numerical Method (ANM) is a powerful continuation method that usually converges faster than Newtonian methods. ANM explores the landscape of the function by following a parameterized solution curve approximated with a high-order power series. Although ANM has

In latest years, several advancements have been made in symbolic-numerical eigenvalue techniques for solving polynomial systems. In this article, we add to this list by reducing the task to an eigenvalue problem in a considerably faster and simpler way than in previous methods. This results in an algorithm which solves systems with isolated solutions in a reliable and efficient way, outperforming homotopy

We extended dynamic time warping (DTW) into interval-based dynamic time warping (iDTW), including (A) interval-based representation (iRep): [1] abstracting raw, time-stamped data into interval-based abstractions, [2] comparison-period scoping, [3] partitioning abstract intervals into a given temporal granularity; (B) interval-based matching (iMatch): matching partitioned, abstract-concepts records

Given a $k\times n$ integer primitive matrix $A$ (i.e., a matrix can be extended to an $n\times n$ unimodular matrix over the integers) with size of entries bounded by $\lambda$, we study the probability that the $m\times n$ matrix extended from $A$ by choosing other $m-k$ vectors uniformly at random from $\{0, 1, \ldots, \lambda-1\}$ is still primitive. We present a complete and rigorous proof that

Let $F(z)$ be an arbitrary complex polynomial. We introduce the local root clustering problem, to compute a set of natural $\varepsilon$-clusters of roots of $F(z)$ in some box region $B_0$ in the complex plane. This may be viewed as an extension of the classical root isolation problem. Our contribution is two-fold: we provide an efficient certified subdivision algorithm for this problem, and we provide

Let $p(x)$ be an integer polynomial with $m\ge 2$ distinct roots $\alpha_1,\ldots,\alpha_m$ whose multiplicities are $\boldsymbol{\mu}=(\mu_1,\ldots,\mu_m)$. We define the D-plus discriminant of $p(x)$ to be $D^+(p):= \prod_{1\le i

As mathematical computing becomes more democratized in high-level languages, high-performance symbolic-numeric systems are necessary for domain scientists and engineers to get the best performance out of their machine without deep knowledge of code optimization. Naturally, users need different term types either to have different algebraic properties for them, or to use efficient data structures. To

We develop a class of algebraic interpretations for many-sorted and higher-order term rewriting systems that takes type information into account. Specifically, base-type terms are mapped to \emph{tuples} of natural numbers and higher-order terms to functions between those tuples. Tuples may carry information relevant to the type; for instance, a term of type $\mathsf{nat}$ may be associated to a pair

Virtualization of computing and networking, IT-OT convergence, cybersecurity and AI-based enhancement of autonomy are significantly increasing the complexity of CPS and CPSoS. New challenges have emerged to demonstrate that these systems are safe and secure. We emphasize the role of control and emerging fields therein, like symbolic control or set-based fault-tolerant and decentralized control, to

Any Hilbert space with composite dimension can be factorized into a tensor product of smaller Hilbert spaces. This allows to decompose a quantum system into subsystems. We propose a simple tractable model for a constructive study of decompositions of quantum systems.

The latest Deep Learning (DL) models for detection and classification have achieved an unprecedented performance over classical machine learning algorithms. However, DL models are black-box methods hard to debug, interpret, and certify. DL alone cannot provide explanations that can be validated by a non technical audience. In contrast, symbolic AI systems that convert concepts into rules or symbols

A phenomenological Hamiltonian of a closed (i.e., unitary) quantum system is assumed to have an $N$ by $N$ real-matrix form composed of a unperturbed diagonal-matrix part $H^{(N)}_0$ and of a tridiagonal-matrix perturbation $\lambda\,W^{(N)}(\lambda)$. The requirement of the unitarity of the evolution of the system (i.e., of the diagonalizability and of the reality of the spectrum) restricts, naturally

We discuss practical methods for computing the space of solutions to an arbitrary homogeneous linear system of partial differential equations with constant coefficients. These rest on the Fundamental Principle of Ehrenpreis-Palamodov from the 1960s. We develop this further using recent advances in computational commutative algebra.

In this paper we study the equations of the elimination ideal associated with $n+1$ generic multihomogeneous polynomials defined over a product of projective spaces of dimension $n$. We first prove a duality property and then make this duality explicit by introducing multigraded Sylvester forms. These results provide a partial generalization of similar properties that are known in the setting of homogeneous

We present a new open source C library msolve dedicated to solving multivariate polynomial systems of dimension zero through computer algebra methods. The core algorithmic framework of msolve relies on Gr{\"o}bner bases and linear algebra based algorithms for polynomial system solving. It relies on Gr{\"o}bner basis computation w.r.t. the degree reverse lexicographical order, Gr{\"o}bner conversion

Verification is one of the central tasks during circuit design. While most of the approaches have exponential worst-case behaviour, in the following techniques are discussed for proving polynomial circuit verification based on Binary Decision Diagrams (BDDs). It is shown that for circuits with specific structural properties, like e.g. tree-like circuits, and circuits based on multiplexers derived from

New algorithms are presented for computing annihilating polynomials of Toeplitz, Hankel, and more generally Toeplitz+ Hankel-like matrices over a field. Our approach follows works on Coppersmith's block Wiedemann method with structured projections, which have been recently successfully applied for computing the bivariate resultant. A first baby-step/giant step approach -- directly derived using known

This work proposes a Prolog-dialect for the found and prioritised problems on expressibility and automation. Given some given C-like program, if dynamic memory is allocated, altered and freed on runtime, then a description of desired dynamic memory is a heap specification. The check of calculated memory state against a given specification is dynamic memory verification. This contribution only considers

Let $\mathbf{f} = \left(f_1, \dots, f_p\right) $ be a polynomial tuple in $\mathbb{Q}[z_1, \dots, z_n]$ and let $d = \max_{1 \leq i \leq p} \deg f_i$. We consider the problem of computing the set of asymptotic critical values of the polynomial mapping, with the assumption that this mapping is dominant, $\mathbf{f}: z \in \mathbb{K}^n \to (f\_1(z), \dots, f\_p(z)) \in \mathbb{K}^p$ where $\mathbb{K}$