Algpath is a certified homotopy continuation software. We upgrade the previous fixed-precision Rust implementation by incorporating mixed, adaptive precision with minimal overhead. This allows us to tackle problems on which the initial implementation fails due to the inability to increase precision, and where uncertified methods may fail or path jump.
When solving systems of polynomial equations and inequalities, the task of computing their solutions with integer coordinates is a much harder problem than that of computing their real solutions or that of computing all their solutions. In fact, in the presence of non-linear constraints, this task may simply become an undecidable problem [12, 15]. However, studying the integer solutions of linear systems of equations and inequalities is of practical importance in various areas of scientific computing. Two such areas are combinatorial optimization (in particular, integer linear programming) and compiler optimization (in particular, the analysis, transformation, and scheduling of nested loops in computer programs), where a variety of algorithms solve questions related to the points with integer coordinates in a given polyhedron. Another area is at the crossroads of computer algebra and polyhedral geometry, with topics such as toric ideals and Hilbert bases, see [16], as well as the manipulation of Laurent series, see [1].
An in-progress pipeline based on B-series is proposed to solve reachability analysis of ordinary differential equations. First, a symbolic step generates a B-series and its associated truncation error. This later is symbolically optimized in order to reduce the over-estimation introduced by computation with sets. Finally, the B-series is evaluated with affine arithmetic and intervals to produce a guaranteed reachable tube.
S -packing coloring is a generalization of proper coloring of graphs, introduced more than a decade ago. In this paper, we present algebraic and combinatorial algorithms for the problem of S -packing coloring of finite undirected and unweighted graphs. We assess the upper bounds for the complexity of our algorithms and of an existing algebraic algorithm by Maarouf [Electron. J. Combin. 2017]. We also provide a comparative evaluation of these algorithms.
We demonstrate the calculation of a number of triple summations, originating from a combinatorics equation. The summations produce Harmonic numbers, and Zeta and Hurwitz zeta functions, as well as prove the equation.
Let E, E′ be two isogenous ordinary elliptic curves defined over a finite field. The set Hom(E, E′) of isogenies from E to E′ is a free ℤ-module of rank two. In this paper, we give an algorithm for finding an explicit ℤ-basis of Hom(E, E′). Firstly, we find a special pair of isogenies using the structure of ordinary isogeny graphs, called isogeny volcanoes. Secondly, we construct explicit generators of Hom(E, E′) as a ℤ-module from the pair of isogenies. Finally, we remove the ℤ-linear dependency of the generators to obtain a ℤ-basis of Hom(E,E′).
We use machine learning to approximate the space of steady states in the shuttle model for the Wnt Signaling Pathway. In this ongoing work, we combine these methods with computational algebra to investigate a question by Gross, Harrington, Rosen, and Sturmfels.
We consider the concept of so-called central path in nonlinear optimization that lies at the heart of interior point methods. Using singularity theory and real algebraic geometry, we provide conditions for the existence, convergence, and analyticity of central paths of nonlinear optimization problems. We also prove quantitative bounds on the convergence rate and analyticity of the central path for the special case of semi-definite optimization.
We present in this extended abstract some ongoing work which has been presented in [13]. This extended abstract contains no new results w.r.t. [13], and is intended to present our prototype integral_elimination for performing Integral Elimination. This prototype is a first step to the generalization of the differential elimination algorithms for the integral case, where the differentiation is replaced by an integration operator. Our approach is incomplete and still needs an important theoretical development. However it is already sufficient to handle nontrivial examples The originality of our approach consists in using new types of reductions and introducing exponential terms during the computations. Our prototype has been implemented in Python using SymPy. We briefly motivate the context, and present some examples treated by our prototype.
The decision problem of the first order theory over the reals is a fundamental problem in symbolic computation. Because many problems emerging from other areas of science and engineering can be directly expressed by a decision problem over the reals, there are more and more interests in developing efficient algorithms for the decision problem. For example, the non-linear real arithmetic (NRA) theory in satisfiability modulo theories (SMT) is a special case of the real decision problem, where all variables are existentially quantified. And there are applications in Robotics, Biology and Economics. More and more new applications are discovered all the time. Besides, the real decision problem is related to many different branches of mathematics, such as logic, geometry, algebra and computation. Therefore, the decision problem over the reals is interesting to both the mathematics community and researchers from other science areas.
We resolve some open conjectures from the OEIS about Hardinian arrays (see A253217). In particular, we show via the transfer matrix method that Hr(n, k), the number of n × k Hardinian arrays with parameter r is a polynomial in n of degree r when r and k are fixed and n is sufficiently large. Our implementation of the main result in Maple and Python involves computing with large, sparse matrices, both numeric and symbolic.
There are a number of ways to generalize the notion of the resultant to non-commutative polynomials, in particular, to quaternionic polynomials. We show that Erić's definition of the resultant admits similar optimization as the standard commutative resultant. Hence, it can be effectively computed.
Let a be a polynomial in Z[x1,x2,…,xn]. Let α ∈ Zn and p be a large prime. Let B be a modular black box representation for a, that is, B : Zn × {p} → Zp such that B(α,p) outputs a(α) mod p. In our implementation B is a Maple procedure. We present a Maple program CMBBSHL which on input of B outputs the irreducible factorization Πri=1 feii of a(x1,…,xn) with high probability. Our program is a combination of Maple codes and C codes where the main programs are implemented in Maple and several subroutines are implemented in C for increased efficiency. We present (1) a description of the algorithm, (2) a demonstration of the software, (3) a timing benchmark, and (4) some implementation details.
Each quarter we are pleased to present abstracts of recent doctoral dissertations in Computer Algebra and Symbolic Computation. We encourage all recent Ph.D. graduates who have defended in the past two years (and their supervisors), to submit their abstracts for publication in CCA. Please send abstracts to the CCA editors for consideration.
We present a new GeoGebra Discovery command, ShowProof, which provides a certified step-by-step algebraic proof and complexity grade of planar geometry theorems. The output is available in multiple languages and can be exported as plain text to Mathematica or Maple format for further analysis.
This paper introduces a Julia package for tackling linear Diophantine systems and related optimization problems using the Polyhedral Omega algorithm. The package integrates partition analysis and polyhedral geometry to offer efficient solutions. PolyhedralOmega.jl package implements the Polyhedral Omega algorithm, and Optimization.jl provides optimization enhancements.
We present a Mathematica package to help investigate the existence of Darboux polynomials for (polynomial) ordinary differential equations (ODE). If we denote the derivation encoded by a given ODE by D, the package infers and propagates necessary conditions for a generic ansatz p to be a Darboux polynomial for D. By generic, we mean that not only the coefficients of p are unknown, but also the exponent d of its leading monomial. We present some examples for which the package produces a formal certificate proving the nonexistence of nontrivial Darboux polynomials. The package complements well standard algorithms that search for Darboux polynomials up to a fixed bound on the total degree.
Chemical Reaction Networks (CRNs) are a standard formalism used in chemistry and biology to model complex molecular interaction systems. In the perspective of systems biology, they are a central tool to analyze the high-level functions of the cell in terms of their low-level molecular interactions. In the perspective of synthetic biology, they constitute a target programming language to implement in chemistry new functions either in vitro, in artificial vesicles, or in living cells. In this paper, we describe the CRN synthesis tool part of our CRN modeling and analysis software BIOCHAM (Biochemical Abstract Machine). This compiler transforms any elementary (resp. algebraic) real function into a formal finite CRN to compute it (resp. with absolute functional robustness), through a pipeline of symbolic computation steps, among which quadratization optimization plays a key role to restrict to elementary reactions with at most two reactants and a minimum number of molecular species.
Each quarter we are pleased to present abstracts of recent doctoral dissertations in Computer Algebra and Symbolic Computation. We encourage all recent Ph.D. graduates who have defended in the past two years (and their supervisors), to submit their abstracts for publication in CCA. Please send abstracts to the CCA editors <[email protected]> for consideration.
