universidade de santiago de compostela · colexio de san xerome pza. obradoiro, s/n 15782 -...

Fifth EACA International School

on Computer Algebra and its Applications

February 25–28, 2020

BCAM, Bilbao, Spain

March 6th, 2018

V.A.T 0,00 €

Total 80,00 €

CIF/FIC: Q-1518001-AColexio de San Xerome

Pza. Obradoiro, s/n15782 - Santiago de Compostela

e-mail: [email protected]

Invoice #

Eduardo Sáenz de Cabezón

Logroño, La Rioja (Spain)VAT: 16.567.689-FInvoice date:

Eur

Universidade de Santiago de Compostela

Final amount

INVOICE

Money Transfer to: IBAN ES40 0049 2584 9022 1400 2210 --- BIC SWIFT BSCHESMMXXX

Fourth EACA International School onComputer Algebra and its ApplicationsSantiago de Compostela, March 20-23, 2018

001/EACA2018

Description Amount

VAT EXEMPTION FOR NON PROFIT ORGANIZATIONS VAT-LAW 49/2002 - DECEMBER 23, 2002

80,00 €

Registration fees of Prof. Eduardo Sáenz de Cabezón 80,00 €

http://www.bcamath.org/es/workshops/5-eaca-school

The school will take place at the Basque Center for Applied Mathematics - BCAM with the support of:

• Basque Center for Applied Mathematics - BCAM http://www.bcamath.org/

• Ministerio de Ciencia e Innovación http://www.ciencia.gob.es/portal/site/MICINN/

Dynamisation Actions “Redes de Investigación” RED2018-102709-T

• Red EACA: Red Temática de Cálculo Simbólico, Álgebra Computacional y Aplicacioneshttp://www.unirioja.es/dptos/dmc/RedEACA/

• Department of Education Gobierno Vascohttp://www.euskadi.eus/basque-government/department-education/

• ikerbasque (Basque Foundation for Science) https://www.ikerbasque.net/

• Bizkaia Diputación Foral https://web.bizkaia.eus/

• Innobasque https://www.innobasque.eus/

• Universidad del País Vasco https://www.ehu.eus/

http://www.bcamath.org/es/workshops/5-eaca-school

http://www.bcamath.org/

http://www.ciencia.gob.es/portal/site/MICINN/

http://www.unirioja.es/dptos/dmc/RedEACA/

http://www.euskadi.eus/basque-government/department-education/

https://www.ikerbasque.net/

https://web.bizkaia.eus/

https://www.innobasque.eus/

https://www.ehu.eus/

Fifth EACA International School BCAM, Bilbao, 2020


on Computer Algebra and its Applications

February 25–28, 2020

BCAM, Bilbao, Spain

Scientific Committee

• María Emilia Alonso (Universidad Complutense de Madrid)

• Enrique Artal (Universidad de Zaragoza)

• Marta Casanellas (Universitat Politècnica de Catalunya)

• Francisco Jesús Castro-Jiménez (Universidad de Sevilla)

• Carlos D’Andrea (Universitat de Barcelona)

• Ignacio García Marco (Universidad de La Laguna)

• Philippe Gimenez (Universidad de Valladolid)

• José Gómez Torrecillas (Universidad de Granada)

• Laureano González Vega (Universidad de Cantabria)

• Manuel Ladra (Universidade de Santiago de Compostela); Chair

• Francisco José Monserrat Delpalillo (Universidad Politécnica de Valencia)

• Sonia Pérez Díaz (Universidad de Alcalá de Henares)

• Ana Romero (Universidad de La Rioja)

Organizing Committee

• Irantzu Elespe (BCAM - Basque Center for Applied Mathematics)

• Javier Fernández de Bobadilla (BCAM - Basque Center for Applied Mathematics)

• Manuel Ladra (Universidade de Santiago de Compostela)

2


Classifying small dimensional algebrasWillem A. De Graaf 6

Code-based Cryptography: an example of Post-quantum CryptographyIrene Márquez CorbellaMarkel Epelde García 7

Algebraic Machine LearningGonzalo G. de PolaviejaFernando Martín Maroto 8

Operator-algebraic elements in prime ringsJose Brox 10

The computation of the b-function of certain holonomic ideals in the Weyl algebraFrancisco J. Castro-Jiménez 11

On Loday-Pirashvili’s categoryAlejandro Fernández-Fariña 12

Anti-self-dual bi-Lagrangian surfacesMaría Ferreiro Subrido 13

Constructive partitioning polynomials and applicationsPablo González Mazón 14

A computational technique to solve a geometrical problemRodrigo Mariño-Villar 15

Morse Theory on finite spacesDavid Mosquera Lois 16

Non-associative central extensions of null-filiform associative algebrasPilar Páez-Guillán 17

Affine equivalences of trigonometric curvesEmily Quintero 18

3


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Speakers

5


Willem A. De Graaf

Università degli Studi di Trento

Classifying small dimensional algebras

https://www.science.unitn.it/~degraaf/

6

https://www.science.unitn.it/~degraaf/


Irene Márquez Corbella

Universidad de La Laguna

Markel Epelde García

Universidad del País Vasco

Code-based Cryptography: an example of Post-quantum Cryptography

http://www.singacom.uva.es/~iremarquez/

7

http://www.singacom.uva.es/~iremarquez/


Gonzalo G. de Polavieja

Fernando Martín Maroto

Champalimaud Foundation, Lisboa, Portugal

Algebraic Machine Learning

https://polaviejalab.org/

8

https://polaviejalab.org/


Contributed talks

9

Operator-algebraic elements in prime rings

Jose Brox

Jose Brox ([email protected])CMUC - Centre for Mathematics, University of Coimbra

Abstract.

Let A be a prime ring with extended centroid C. An element a ∈ A is nilpotent if there is n ∈ Nsuch that an = 0. Analogously, considering the Lie ring A− with bracket product [x, y] := xy − yxand ad : A→ A operator adxy := [x, y], we say that an element a ∈ A is ad-nilpotent if there is n ∈ Nsuch that adna = 0 (as a map). For example, ad1a = 0 if and only if [a,R] = 0, i.e., iff a ∈ Z(A) iscentral, and ad2a = 0 if and only if a2x − 2axa + xa2 = 0 for all x ∈ A. By a theorem of Martindaleand Miers, an element is ad-nilpotent if and only if it is the sum of a nilpotent and a central element,so it is algebraic with minimal polynomial (X − λ)m ∈ C[X] for some λ ∈ C and m ∈ N. Morein general, for a ∈ A let La, Ra denote the left and right multiplicator operators respectively (i.e.,Lax := ax,Rax := xa) and from any fixed polynomial in 2 variables f ∈ C[X,Y ] define the operatorfa := f(La, Ra) for each a ∈ A. For example, if f(X,Y ) = X2−2XY +Y 2 then fa = L2

a−2LaRa+R2a

and fa(x) = a2x− 2axa+ xa2 = ad2a(x). We say that the element a ∈ A is f -algebraic if fa = 0 as amap. Generalizing Martindale and Miers theorem to its fullest, we show that any f -algebraic elementis algebraic for any f , and find the possible minimal polynomials that the f -algebraic elements canpresent for a given f . To do so we entirely rewrite the problem as a question in ideals of polynomialrings in two variables and apply basic theory of Gröbner bases, the partial Hasse derivatives, and amultiset form of Alon’s combinatorial nullstellensatz.


10

The computation of the b-function of certain holonomic idealsin the Weyl algebra

Francisco J. Castro-Jiménez

F. J. Castro-Jiménez ([email protected])Universidad de Sevilla

Abstract.

We explain how to apply Noro’s algorithm to compute the Bernstein-Sato polynomial bI,ω(s) whenI belongs to a family of hypergeometric ideals in the Weyl algebra and ω is a real weight vector.

This is a joint work with Helena Cobo.


11

On Loday-Pirashvili’s category

Alejandro Fernández-Fariña

A. Fernández-Fariña ([email protected])Universidade de Santiago de Compostela

Abstract.

In [1] Loday and Pirashvili equipped the category LM of linear maps of vector spaces with a tensorproduct, constructing Loday-Pirashvili’s category. This tensor product allows them to do variousconstructions related to Leibniz algebras; in this monoidal category, a Leibniz algebra becomes a Lieobject. In this talk, we will introduce this category and show that its construction can be done forcategories other than vector spaces asking for some conditions that are not very restrictive.

References[1] J.-L. Loday, T. Pirashvili. The tensor category of linear maps and Leibniz algebras. Georgian

Math. J. 5 (1998), 263–276.


12

Anti-self-dual bi-Lagrangian surfaces

María Ferreiro Subrido

María Ferreiro Subrido ([email protected])Universidade de Santiago de Compostela

Abstract.

The aim of this talk is to introduce and give the local description of anti-self-dual bi-Lagrangiansurfaces. Noticing that the underlying structure of a bi-Lagrangian surface is that of a Walker 4-manifold, we will see that an anti-self-dual bi-Lagrangian surface is locally isometric to the cotangentbundle of an affine surface equipped with a modified Riemannian extension given by a vector field X,a (1, 1)-tensor field T and a symmetric (0, 2)-tensor field Φ on the affine surface.

References[1] B. Bryant. Bochner-Kähler metrics. J. Amer. Math. Soc. 14 (2001), 623–715.

[2] E. Calviño-Louzao, E. García-Río, P. Gilkey, R. Vázquez-Lorenzo. The geometry of modifiedRiemannian extensions. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 45 (2009), 2023–2040.

[3] E. Calviño-Louzao, E. García-Río, P. Gilkey, I. Gutiérrez-Rodríguez, R. Vázquez-Lorenzo. Affinesurfaces which are Kähler, para-Kähler, or nilpotent Kähler. Results Math. 73 (4) (2018), Art.135, 24 pp.

[4] A. G. Walker. Canonical form for a Riemannian space with a parallel field of null planes. Quart.J. Math. Oxford Ser. (2) 1 (1950), 69–79.


13

Constructive partitioning polynomials and applications

Pablo González Mazón

Pablo González Mazón ([email protected])Université Gustave EiffelLabex Bézout fellow

Abstract.

The talk is an introduction to the state-of-the-art of partitioning polynomials, focusing on con-structive strategies. Both theoretical and applied results are discussed, but special emphasis is put onthe algorithmic difficulties and how these have been tackled to this date. Most relevant applications ofpartitioning polynomials to computer science are reviewed, as well as other possible new approaches.

References[1] P. K. Agarwal, B. Aronov, E. Ezra, J. Zahl. An efficient algorithm for generalized polynomial

partitioning and its applications. 35th International Symposium on Computational Geometry,Art. No. 5, 14 pp., LIPIcs. Leibniz Int. Proc. Inform., 129, Schloss Dagstuhl. Leibniz-Zent. Inform.,Wadern, 2019.

[2] P. K. Agarwal, J. Matousek. On range searching with semialgebraic sets (II). Discrete andComputational Geometry 11 (4) (1994), 393–418.

[3] S. Basu, R. Pollack, M.-F. Roy. Algorithms in real algebraic geometry. Second edition. Algorithmsand Computation in Mathematics, 10. Springer-Verlag, Berlin, 2006

[4] L. Guth, N. H. Katz. On the Erdös distinct distances problem in the plane Ann. of Math. (2)181 (1) (2015), 155–190.

[5] L. Guth. Polynomial partitioning for a set of varieties. In Mathematical Proceedings of the Cam-bridge Philosophical Society 159 (2015), 459–469.

[6] C. Knauer, H. R. Tiwary, D. Werner. On the computational complexity of ham-sandwich cuts,Helly sets, and related problems. 28th International Symposium on Theoretical Aspects of Com-puter Science, 649–660, LIPIcs. Leibniz Int. Proc. Inform., 9, Schloss Dagstuhl. Leibniz-Zent.Inform., Wadern, 2011.

[7] C.-Y. Lo, J. Matousek, W. Steiger. Algorithms for ham-sandwich cuts. Discrete and ComputationalGeometry 11 (4) (1994), 433–452.

[8] M. N. Walsh. Polynomial partitioning over varieties. arXiv:1811.07865 (2018).


14

A computational technique to solve a geometrical problem

Rodrigo Mariño-Villar

Rodrigo Mariño Villar ([email protected])Universidade de Santiago de Compostela

Abstract.

Einstein metrics are critical metrics of the Hilbert-Einstein functional g 7→∫Mτgdvolg when re-

stricted to metrics of constant volume in dimension n ≥ 3. This is a subject that has been studied inlots of texts and with different points of view and fields in both mathematics and physics.

In dimension four, the Gauss-Bonnet integrand ‖R‖2−4‖ρ‖2+τ2 gives rise to a universal curvatureidentity [2]

(R− ‖R‖

2

4g

)+ τ

(ρ− τ

4g)− 2

(ρ− ‖ρ‖

2

4g

)− 2

(R[ρ]− ‖ρ‖

2

4g

)= 0,

where R, ρ and R[ρ] are the symmetric (0, 2)-tensor fields given by Rij = RiabcRjabc ρij = ρiaρ

aj

and R[ρ]ij = Riabjρab. The tensor fields R, ρ and R[ρ] provide natural Riemannian invariants,

algebraically the simplest ones after the Ricci tensor, which has not received much attention in theliterature.

We say that a four-dimensional Riemannian manifold (M, g) is weakly-Einstein if it satisfies oneof the following conditions:

(M, g) is R-Einstein if R = 14‖R‖2g but (M, g) is not Einstein. Similarly (M, g) is said to be

ρ-Einstein (resp., R[ρ]-Einstein) if ρ = 14‖ρ‖2g (resp., R[ρ] = 1

4‖ρ‖2g) and (M, g) is not Einstein.The aim of this talk is to provide a classification of these three conditions in the four dimensional

homogenous setting, where all possible manifolds are know by [1] . In order to do this, we are usingthe powerful tool of Gröbner basis to simplify systems of non-linear polynomials.

References[1] L. Bérard-Bergery, Les spaces homogènes Riemanniens de dimension 4. Riemannian geometry in

dimension 4 (Paris 1978/1979) 3 (1981), 40–60.

[2] Y. Euh, J. Park, K. Sekigawa, A curvature identity on a 4-dimensional Riemannian manifold.Result. Math. 63 (2013), 107–114


15

Morse Theory on finite spaces

David Mosquera Lois

D. Fernández-Ternero ([email protected])Universidad de Sevilla

E. Macías-Virgós ([email protected])Universidade de Santiago de Compostela

D. Mosquera Lois ([email protected])Universidade de Santiago de Compostela

N.A. Scoville ([email protected])Ursinus College (USA)

J.A. Vilches ([email protected])Universidad de Sevilla

Abstract.

Morse Theory was developed in the twenties as a tool to study the topology (homology, originally)of a manifold by breaking it into “elementary” pieces. In the nineties, R. Forman introduced a combi-natorial approach to Morse Theory in the context of simplicial or regular CW-complexes, now referredas Discrete Morse Theory [2]. The aim of this talk is to motivate the importance of the study of finitetopological spaces and to present a computational Morse Theory in this context [1, 3].

References[1] D. Fernández-Ternero, E. Macías-Virgós, D. Mosquera-Lois, N.A. Scoville, J.A. Vilches. Funda-

mental Theorems of Morse Theory on posets. Preprint (2020), 24 pp.

[2] R. Forman. Morse theory for cell complexes. Adv. Math. 134 (1) (1998), 90–145.

[3] E. G. Minian. Some remarks on Morse theory for posets, homological Morse theory and finitemanifolds. Topology Appl. 159 (12) (2012), 2860–2869.


16

Non-associative central extensions of null-filiform associativealgebras

Pilar Páez-Guillán

Pilar Páez-Guillán ([email protected])Universidad de Santiago de Compostela

Abstract.

The algebraic study of central extensions of different varieties of non-associative algebras playsan important role in the classification problem in such varieties. Since Skjelbred and Sund in 1978devised a method for classifying nilpotent Lie algebras, making crucial use of central extensions, it hasbeen adapted to many other varieties of algebras, including associative, Malcev, Jordan, and manyothers.

The study of central extensions of null-filiform algebras was initiated in [2] within the frameworkof Leibniz algebras: all Leibniz central extensions of null-filiform Leibniz algebras were described.More recently [1], it was proven that there is only one associative central extension of the associativenull-filiform algebra µn

0 ; namely, µn+10 . However, null-filiform algebras can be considered as elements

of more general varieties of algebras, such as alternative, left alternative, Jordan, bicommutative, leftcommutative, assosymmetric, Novikov or left symmetric, among others. In this work, we classifythe isomorphism classes of central extensions of the associative null-filiform algebra µn

0 over severalvarieties of non-associative algebras, as the ones mentioned above.

This is a joint work with Ivan Kaygorodov and Samuel Lopes.

References[1] J. Adashev, L. Camacho, B. Omirov. Central extensions of null-filiform and naturally graded

filiform non-Lie Leibniz algebras. Journal of Algebra 479 (2017), 461–486.

[2] I. Karimjanov, I. Kaygorodov, M. Ladra. Central extensions of filiform associative algebras. Linearand Multilinear Algebra (2019). doi: 10.1080/03081087.2019.1620674.


17

Affine equivalences of trigonometric curves

Emily Quintero

Emily Quintero ([email protected])Universidad de Alcalá

Juan Gerardo Alcázar ([email protected])Universidad de Alcalá

Abstract.

We provide an efficient algorithm to detect whether two given trigonometric curves, i.e. twoparametrized curves whose components are truncated Fourier series, in any dimension, are affinelyequivalent, i.e. whether there exists an affine mapping transforming one of the curves onto the other.If the coefficients of the parametrizations are known exactly (the exact case), the algorithm boils downto univariate gcd computation, so it is efficient and fast. If the coefficients of the parametrizationsare known with finite precision, e.g. floating point numbers (the approximate case), the univariategcd computation is replaced by the computation of singular values of an appropriate matrix. Ourexperiments show that the method works well, even for high degrees.

References[1] Alcázar J.G., Hermoso C., Muntingh G. Detecting similarity of Rational Plane Curves. Journal of

Computational and Applied Mathematics 269 (2014), 1–13.

[2] Hauer M., Jüttler B. Projective and affine symmetries and equivalences of rational curves inarbitrary dimension. Journal of Symbolic Computation 87 (2018), 68–86.

[3] Hauer M., Jüttler B., Schicho J. Projective and affine symmetries and equivalences of rational andpolynomial surfaces. Journal of Computational and Applied Mathematics (2018), in Press.

[4] Hong H. Implicitization of curves parametrized by generalized trigonometric polynomials. Proceed-ings of Applied Algebra, Algebraic Algorithms and Error Correcting Codes (AAECC-11) (1995),pp. 285–296.

[5] Hong H., Schicho J. Algorithms for Trigonometric Curves (Simplification, Implicitization, Param-eterization). Journal of Symbolic Computation 26 (3) (1998), 279–300.

[6] Karmarkar N., Lakshman Y. N. On Approximate GCDs of Univariate Polynomials. Journal ofSymbolic Computation 26 (6) (1998), 653–666.


18

[7] Noda M.T., Sasaki T. Approximate GCD and its application to ill-conditioned equations. Journalof Computational and Applied Mathematics 38 (1991), 335–351.

[8] Sendra J.R., Winkler F., Pérez-Díaz S. Rational Algebraic Curves. Springer-Verlag, 2008.

[9] Stewart, G. W. Perturbation of the SVD in the presence of small singular values. Linear Algebraand its Applications 419 (2006), 53–77.


19


List of participants

Alquézar Baeta, Carlos Universidad de Zaragoza

Angulo, Jorge Universidad de Valladolid

Brox López, José Ramón CMUC (Centre for Mathematics of the University of Coimbra)

Castro Jiménez, Francisco J. Universidad de Sevilla

Coltraro, Franco Instituto de Robótica e Informática Industrial (CSIC-UPC)

De Graaf, Willem A. Università degli Studi di Trento

De Polavieja, Gonzalo G. Champalimaud Foundation, Lisboa

del Río Almajano, Miguel Tereso Universidad de Valladolid

Díez García, Sergio Universidad de Valladolid

Epelde García, Markel Universidad del País Vasco

Fernández de Bobadilla, Javier BCAM - Basque Center for Applied Mathematics

Fernández Fariña, Alejandro Universidade de Santiago de Compostela

Ferreiro Subrido, María Universidade de Santiago de Compostela

Galindo, Guillermo Universidad de Granada

García Martínez, Xabier Universidade de Vigo

Gimenez, Philippe Universidad de Valladolid

González Mazón, Pablo Université Gustave Eiffel

Gutiérrez Rodríguez, Ixchel D. Universidade de Vigo

Ladra, Manuel Universidade de Santiago de Compostela

Marín Aragón, Daniel Universidad de Cádiz

Mariño Villar, Rodrigo Universidade de Santiago de Compostela

Márquez Corbella, Irene Universidad de La Laguna

Martín Maroto, Fernando Champalimaud Foundation, Lisboa

Mosquera Lois, David Universidade de Santiago de Compostela

Pérez Callejo, Elvira Universidad de Valladolid

Pérez Guillán, Pilar Universidade de Santiago de Compostela

Puig i Surroca, Gil Universitat de Barcelona

Quintero, Emily Universidad de Valladolid

Samperio Valdivieso, Álvaro Fundación CARTIF, Valladolid

San José, Rodrigo Universidad de Valladolid

Segovia Martín, José Ignacio Universidad de Valladolid

Señas Peón, Pablo Universidad de Cantabria

20


Sponsors

• Basque Center for Applied Mathematics - BCAM http://www.bcamath.org/

• Ministerio de Ciencia e Innovación http://www.ciencia.gob.es/portal/site/MICINN/

“Redes de Excelencia” MTM2016-81932REDT

• Red EACA: Red Temática de Cálculo Simbólico, Álgebra Computacional y Aplicacioneshttp://www.unirioja.es/dptos/dmc/RedEACA/

• Department of Education Gobierno Vascohttp://www.euskadi.eus/basque-government/department-education/

• ikerbasque (Basque Foundation for Science) https://www.ikerbasque.net/

21

http://www.bcamath.org/

http://www.ciencia.gob.es/portal/site/MICINN/

http://www.unirioja.es/dptos/dmc/RedEACA/

http://www.euskadi.eus/basque-government/department-education/

https://www.ikerbasque.net/


• Bizkaia Diputación Foral https://web.bizkaia.eus/

• Innobasque https://www.innobasque.eus/

• Universidad del País Vasco https://www.ehu.eus/

22

https://web.bizkaia.eus/

https://www.innobasque.eus/

https://www.ehu.eus/

universidade de santiago de compostela · colexio de san xerome pza. obradoiro, s/n 15782 -...

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