Session abstracts

Session S24 - Symbolic Computation: Theory, Algorithms and Applications



Tuesday, July 13, 12:00 ~ 12:25 UTC-3

Newton's Lemma for differential equations

Fuensanta Aroca

Universidad Nacional Autónoma de México, México   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

The Newton method for plane algebraic curves is based on the following remark: the first term of a series, root of a polynomial with coefficients in the ring of series in one variable, is a solution of an initial equation that can be determined by the Newton polygon. Given a monomial ordering in the ring of polynomials in several variables, we describe the systems of initial equations that satisfy the first terms of the solutions of a system of partial differential equations. As a consequence, we extend Mora and Robbiano’s Groebner fan to differential ideals.

Joint work with Giovanna Ilardi (Universidad de Nápoles).

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Tuesday, July 13, 12:30 ~ 12:55 UTC-3

On the Fundamental Theorem of Tropical Partial Differential Algebraic Geometry

Sebastian Falkensteiner

RISC Hagenberg, Johannes Kepler University Linz, Austria   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Tropical Differential Algebraic Geometry considers difficult or even intractable problems in Differential Equations and tries to extract information on their solutions from a restricted structure of the input. The Fundamental Theorem of Tropical Differential Algebraic Geometry and its extensions state that the support of power series solutions of systems of partial differential equations can be obtained either, by solving a so-called tropicalized differential system, or by testing monomial-freeness of the associated initial ideals instead of analyzing the given system itself. Tropicalized differential equations work on a very simple algebraic structure which may help in theoretical and computational questions, particularly on the existence of solutions.

The content of the talk will be the introduction of the underlying algebraic structures, and the presentation of the precise statement of the Fundamental Theorem and the latest results on its extension and generalization.

Joint work with Cristhian Garay-Lopez (Centro de Investigacion en Matematicas, Mexico), Mercedes Haiech (Institut de recherche mathematique de Rennes, France), Marc Paul Noordman (Bernoulli Institute, University of Groningen, Netherlands) and Francois Boulier (Univ. Lille, CNRS, France).

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Tuesday, July 13, 13:00 ~ 13:25 UTC-3

Initial degenerations for systems of algebraic differential equations

Cristhian Garay López

Center for Research in Mathematics (CIMAT), México   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

By a degeneration, we mean a process that transforms a geometric object $X/F$ defined over a field $F$ into a simpler object that retains many of the relevant properties of $X$. Formally, any degeneration is realized by an integral model for $X$; that is, a flat scheme $X’/R$ defined over some integral domain $R$ whose generic fiber is the original object $X$.

In this talk, we endow the field $F=K(\!(t_1,\ldots,t_m)\!)$ of quotients of  multivariate formal power series with a  generalized non-Archimedean absolute value $|\cdot|$, and we establish the existence of integral models over the  ring of integers $R=\{|x|\leq 1\}$ for solutions $X$ of systems of algebraic partial differential equations with coefficients on $F$. We also concretely describe the specialization map of a model $X’/R$ to the maximal ideals of $R$, which are encoded in terms of monomial orderings.

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Tuesday, July 13, 14:00 ~ 14:25 UTC-3

Value set of Fractional Ideals and the Analytic Classification of Plane Curves

Marcelo Hernandes

Universidade Estadual de Maringá, Brazil   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk we present a solution to the problem of the analytic classification of plane curves with several irreducible components. In our approach the value set of a fractional ideal of the local ring of a plane curve plays a central role in our main result, that is, normal forms for analytic plane curves.

Joint work with Maria Elenice Rodrigues Hernandes (Universidade Estadual de Maringá, Brazil).

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Tuesday, July 13, 14:30 ~ 14:55 UTC-3

DD-finite functions: working beyond holonomic

Antonio Jiménez-Pastor

Johannes Kepler University Linz, Research Institute for Symbolic Computation, Austria   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Holnomic (or D-finite) functions are a well known class of formal power series that satisfy a linear differential equation with polynomial coefficients. This representation, with differential equation plus some initial values, requires only a finite amount of data.

In this talk we present the newer class of DD-finite functions, i.e., formal power series that satisfy a linear differential equation with D-finite coefficients. This new class satisfy plenty of the closure properties that also hold for holonomic functions. We show here some of the algebraic properties that have been proven for DD-finite functions and also a Sage package that allow to manipulate these DD-finite functions and allow the user to prove identities automatically.

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Tuesday, July 13, 15:00 ~ 15:25 UTC-3

Multistationarity and structure in enzymatic networks

Mercedes Perez Millan

UBA-CONICET, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

In this work we focus on structured enzymatic networks modeled under mass-action kinetics [Dickenstein-Perez Millan-Shiu-Tang(2019), Perez Millan-Dickenstein(2018)] and study their regions of multistationarity in parameter space, and also the minimal number of intermediate species and their location in the network to allow for more than one steady state. These questions have been studied in [Feliu-Wiuf(2013)] and also in [Sadeghimanesh-Feliu(2019)] using degree theory techniques. Our approach significantly simplifies the analysis to determine whether the system is multistationary or not. We apply our results on several biologically relevant signaling networks with the aid of computer algebra systems.

Joint work with Alicia Dickenstein (UBA-CONICET, Argentina), Magalí Giaroli (UBA, Argentina) and Rick Rischter (UNIFEI, Brazil).

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Tuesday, July 13, 16:00 ~ 16:25 UTC-3

Spectral factorization of algebro-geometric differential operators

Sonia L. Rueda

Universidad Politécnica de Madrid, Spain   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

In 1923, J.L. Burchnall and T.W. Chaundy established a correspondence between commuting differential operators and algebraic curves. It was already known (I. Schur, 1904) that the centralizer of an ordinary differential operator $L$ has a quotient field that is the function field of one variable. Therefore centralizers can be seen as the affine rings of curves, and in a formal sense these are spectral curves.

The theory of commuting differential operators is well developed for commuting ordinary differential operators. However, for rank greater than one (the rank being the greatest common divisor of the orders of all the elements in its centralizer) or for special spectral curves, this theory is not complete enough and continues to evolve. It has broad connections with many branches of modern mathematics, first of all with integrable systems, since explicit examples of commuting operators provide explicit solutions of many non-linear partial differential equations.

Algebro-geometric ordinary differential operators are defined to have nontrivial centralizers. The spectral curve of an algebro-geometric Schrödinger operator $L$ is a plane algebraic curve whose defining equation $f(\lambda,\mu)=0$ can be computed by means of the differential resultant (E. Previato, 1991). The coefficients of $L$ belonging to a differential field $K$, whose field of constants $C$ is algebraically closed and of characteristic zero. In this talk, we present a symbolic algorithm for the factorization of an algebro-geometric Schrödinger operator $L-\lambda$ over the field $K(\Gamma)$ of its spectral curve $\Gamma$, using differential subresultants [1]. This is what we call a "spectral factorization" and our ultimate goal is an effective approach to the direct spectral problem. Since the field of constants $C(\Gamma)$ of $K(\Gamma)$ is no longer algebraically closed, the need arises of a new algebraic structure, generated by the solutions of the spectral problem $L\Psi=\lambda\Psi$ over $\Gamma$ called "Spectral Picard-Vessiot field" of $L-\lambda$, defined in [2]. The spectral parameter $\lambda$ is not a free parameter. Restricting to the case of rational spectral curves, we transform the original spectral problem to one-parameter form, where the field of the curve is now $C(\tau)$, with a free parameter $\tau$.

We extended our symbolic algorithm to the factorization of a third order algebro-geometric ordinary differential operator $L-\lambda$ [4]. In this context, the first example of a non-planar spectral curve arises. The spectral factorization over planar spectral curves in the case of algebro-geometric operators of fourth order with rank $2$, see [3], is a complicated problem; we will show how the spectral factorization is affected by the rank of the operator.


[1] J.J. Morales-Ruiz. S.L. Rueda, and M.A. Zurro. Factorization of KdV Schr\" odinger operators using differential subresultants. Adv. Appl. Math., 120:102065, 2020.

[2] J.J. Morales-Ruiz. S.L. Rueda, and M.A. Zurro. Spectral Picard-Vessiot fields for algebro-geometric Schrödinger operators . To appear in Ann. Inst. Fourier. See arXiv:1708.00431v3, 2021.

[3] E. Previato, S.L. Rueda, and M.A. Zurro. Commuting Ordinary Differential Operators and the Dixmier Test. SIGMA Symmetry Integrability Geom. Methods Appl., 15(101):23 pp., 2019.

[4] S.L. Rueda and M.A. Zurro. Factoring Third Order Ordinary Differential Operators over Spectral Curves. See arXiv:2102.04733v1, 2021.

Joint work with Maria-Angeles Zurro (Universidad Autónoma de Madrid), Juan J. Morales-Ruiz (Universidad Politécnica de Madrid) and Emma Previato (Boston University).

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Tuesday, July 13, 16:30 ~ 16:55 UTC-3

Walsh-Hadamard transforms of generalized $p$-ary functions and $C$-finite sequences

Luis Medina

University of Puerto Rico, Rio Piedras, Puerto Rico   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk we show that Walsh-Hadamard transformations of generalized $p$-ary functions whose components are symmetric, rotation symmetric or a combination or concatenation of them are $C$-finite sequences. This result generalized many of the known results for regular $p$-ary functions. We also present a study of the roots of the characteristic polynomials related to these sequences and show that properties like balancedness and being bent are not shared by the underline $p$-ary functions.

Joint work with Leonid B. Sepúlveda (University of Puerto Rico, Rio Piedras) and César Serna-Rapello (University of Puerto Rico, Rio Piedras).

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Tuesday, July 13, 17:00 ~ 17:25 UTC-3

A parametric version of the LLL algorithm and consequences

Tristram Bogart

Universidad de los Andes, Colombia   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Given a parametric lattice with a basis given by polynomials with integer coefficients, we give an algorithm to construct an LLL-reduced basis whose elements are eventually quasi-polynomial in t: that is, they are given by formulas that are piecewise polynomial in t (for sufficiently large t), such that each piece is given by a congruence class modulo a period. As a consequence, we show that there are parametric solutions of the shortest vector problem (SVP) and closest vector problem (CVP) that are also eventually quasi-polynomial in t.

Joint work with John Goodrick (Universidad de los Andes, Colombia) and Kevin Woods (Oberlin College, USA).

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Tuesday, July 13, 17:30 ~ 17:55 UTC-3

On positivity of holonomic sequences

Veronika Pillwein

RISC, Joh. Kepler Univ. Linz, Austria   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Holonomic sequences are very well-behaved obects: they satisfy linear recurrences with polynomial coefficients and together with sufficiently many initial values, this gives an exact and finite representation. Over the past decades, several algorithms have been developed and implemented to automatically derive and prove identitities on holonomic sequences. Proving inequalities involving such expressions can not yet be done in such a systematic automatized way. Still, there are some recent methods and structural results for inequalities on holonomic sequences. In this talk, we give an overview on some of these results and show some successful applications of existing software.

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Tuesday, July 20, 16:00 ~ 16:25 UTC-3

Mahler residues and telescopers for rational functions

Carlos Arreche

The University of Texas at Dallas, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

We develop a notion of Mahler discrete residues for rational functions, with the desired property that a given rational function $f(x)$ is of the form $g(x^p)-g(x)$ for some rational function $g(x)$ (where $p$ is an integer $\geq 2$) if and only if all of its Mahler discrete residues vanish. We also show how to apply the technology of Mahler discrete residues to creative telescoping problems. This work extends to the Mahler case the earlier analogous notions, properties, and applications of discrete residues (in the shift case) and $q$-discrete residues (in the $q$-difference case) developed by Chen and Singer.

Joint work with Yi Zhang (Xi'an Jiaotong-Liverpool University, China).

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Tuesday, July 20, 16:30 ~ 16:55 UTC-3

A New Approach to the Connection Problem Using Analytic Combinatorics in Several Variables

Stephen Melczer

University of Waterloo, Canada   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

A foundational problem in enumerative and analytic combinatorics asks one to take an integer sequence defined by some kind of recurrence and determine its dominant asymptotic behaviour. For sequences satisfying linear recurrences with polynomial coefficients, an asymptotic decomposition can often be deduced as a $\mathbb{C}$-linear combination of a finite set of explicit functions. Unfortunately, there is currently no known algorithm to determine these "connection" coefficients, or even decide which are non-zero. In this talk we describe a new approach to compute connection coefficients for sequences arising as "diagonals" of multivariate rational functions. Our approach combines new Morse-theoretic tools from the study of analytic combinatorics in several variables (ACSV) with computer algebra methods for rigorous numeric analytic continuation of functions satisfying linear ODEs. We illustrate this approach on our motivating application, a series of positivity conjectures of Straub and Zudilin.

Joint work with Yuliy Baryshnikov (University of Illinois, Urbana-Champaign) and Robin Pemantle (University of Pennsylvania).

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Tuesday, July 20, 17:00 ~ 17:25 UTC-3

A sage package for n-gonal equisymmetric stratification of $\mathcal{M}_g$

Anita Rojas

Universidad de Chile, Chile   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk we will present an algorithm running over the software SAGE, which allows users to deal with group actions on Riemann surfaces up to topological equivalence. Our algorithm allows us to study the equisymmetric stratification of the branch locus $\mathcal{B}_g$ of the moduli space $\mathcal{M}_g$ of compact Riemann surfaces of genus $g\geq 2$, corresponding to group actions with orbit genus $0$.

Our approach is towards studying inclusions and intersections of (closed) strata of $\mathcal{B}_g$. We apply our algorithm to describe part of the geometry of the branch locus $\mathcal{B}_9$, in terms of equisymmetric stratification. We also use it to compute all group actions up to topological equivalence for genus $5$ to $10$, this completes the lists existing in the literature. Finally, we add an optimized version of a known algorithm, which allows us to identify Jacobian varieties of CM-type. As a byproduct, we obtain a Jacobian variety of dimension $11$ which is isogenous to $E_i^{9}\times E_{i\sqrt{3}}^{2}$, where $E_i$ and $E_{i\sqrt{3}}$ are elliptic curves with complex multiplication.

Joint work with Antonio Behn (Pontificia Universidad Católica de Chile) and Miguel Tello-Carrera (Colegio Pedro de Valdivia).

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Tuesday, July 20, 18:00 ~ 18:25 UTC-3

Univariate Rational Sum of Squares

Agnes Szanto

North Carolina State University, United States   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

It is well-known that a non-negative univariate real polynomial is a sum of 2 squares of real polynomials. Landau in 1905 proved that every univariate polynomial with rational coefficients which is strictly positive on R is a sum of 8 squares of rational polynomials. However, for rational polynomials that are non-negative on R, Scheiderer in 2013 constructed examples which are not sums of squares of rational polynomials. In this talk we consider the local case, namely, polynomials that are non-negative on the real roots of another non-zero polynomial. Parrilo in 2003 gave a simple construction that implies that if f in R[x] is square-free and g in R[x] is non-negative on the real roots of f then g is a sum of squares of real polynomials modulo f. We extend this result to the case when f is an arbitrary rational polynomial and g in Q[x] is non-negative on the real roots of f, assuming that gcd(f/d,d)=1 for d=gcd(f,g). In this case we prove that g is a positive rational combination of squares of rational polynomials modulo f. Moreover, we give bounds on the size of such rational sum of square decomposition and compare algorithmic approaches to compute one.

Joint work with Teresa Krick (Universidad de Buenos Aires) and Bernard Mourrain (INRIA, Sophia Antipolis).

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Tuesday, July 20, 18:30 ~ 18:55 UTC-3

A deflation for bivariate Groebner bases

Eric Schost

University of Waterloo, Canada   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Conca and Valla gave an explicit parametrization of the so-called Groebner cell of all bivariate ideals sharing the same lexicographic leading term ideal. We show how to use this description to design a Newton iteration for bivariate ideals that works in the presence of multiplicities.

Joint work with Catherine St-Pierre (University of Waterloo).

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Tuesday, July 20, 19:00 ~ 19:25 UTC-3

Strictly positive polynomials in the boundary of the SOS cone

Santiago Laplagne

Universidad de Buenos Aires, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk we study the boundary of the cone of real polynomials that can be decomposed as a sum of squares (SOS) of real polynomials. This cone is included in the cone of nonnegative polynomials and both cones share a part of their boundary, which corresponds to polynomials that vanish at at least one point. We focus on the part of the boundary which is not shared, corresponding to strictly positive polynomials.

For the cases of polynomials of degree 6 in 3 variables and degree 4 in 4 variables, this boundary has been completely characterized by G. Blekherman. For the cases of more variables or higher degree, results by G. Blekherman, R. Sinn and M. Velasco and other authors based on general conjectures give bounds for the maximum number of polynomials that can appear in a SOS decomposition and the maximum rank of the matrices in the Gram spectrahedron. However very few concrete examples are known and hence in many cases it is not possible to determine the optimality of the bounds.

We show that these bounds can also be deduced from a conjecture by D. Eisenbud, M. Green and J. Harris. Combining theoretical results and computational techniques, we find new examples and counterexamples that allow us to prove the optimality of the bound in several cases and better understand which result for the cases studied by G. Blekherman can be extended to the general case.

Joint work with Marcelo Valdettaro (Universidad de Buenos Aires).

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Guacarí Modal automatic demonstrator for $K$ and $S2$

Luz Amparo Carranza Guerrero

Fundación Universitaria Konrad Lorenz, Colombia   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

\small Modal logic is an extension of classical logic in which modal operators such as $\square$ for necessity and $\lozenge$ for possibility have been introduced. In 1912 C. I. Lewis defined a new conditional called strict implication; this is defined in terms of the material conditional and the notion of necessity $\square(p\rightarrow q)$. The combinations of the axioms proposed by Lewis provided different normal logics, among which \textit{K, T, B, S4} and \textit{S5} are the most used ones.\\

Non-normal logics, also known as Lewis \textit{S1, S2} and \textit{S3} systems, as well as some of their extensions, \textit{S6, S7, S8} and \textit{S9}, share the characteristics of having non-normal worlds, i.e. worlds lacking formulas with the operator of necessity. These extensions were presented in 1943 by Alban, who extends \textit{S2} with the axiom $\lozenge \lozenge p$. In 1963 and 1965 Saul Kripke published ``Semantical Analysis of Modal Logic'', ``Semantical Analysis of Modal Logic II: Non-Normal Propositional Calculi'', which provided a basic tool for semantical analysis of modal logic and non-normal modal systems.\\

From Kripke's semantical perspective, an effective but not very dynamic method is presented for different\-iang valid and invalid formulas based on Beth's analytical tableaux. The analytical tableaux have been widely used in the calculus of non classical logics, because they facilitate the carrying out of proofs and they have favorable algorithmic and semantical features. Non-normal logics are semantically character\-ized by admitting non-normal worlds in their models, i.e. worlds in which the usual semantical conditions for modal operators are not valid.\\

This paper presents the algorithm for the calculus of tableaux for the \textit{K} and \textit{S2} logics, with characterization of non-normal worlds using the Smullyan tree method. In addition, the Guacarí Modal algorithm will be shown. This automatic demonstrator provides users with a demonstration of the evaluated formula or a refutation of the formula in the above-mentioned systems.

Joint work with Luz Amparo Carranza G, Juan Camilo Acosta, (Fundación Universitaria Konrad Lorenz).

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