The sl3 Jones polynomial of the trefoil: A case study of q-holonomic sequences

AbstractThe sl3 colored Jones polynomial of the trefoil knot is a q-holonomic sequence of two variables with natural origin, namely quantum topology. The paper presents an explicit set of generators for the annihilator ideal of this q-holonomic sequence as a case study. On the one hand, our results are new and useful to quantum topology: this is the first example of a rank 2 Lie algebra computation concerning the colored Jones polynomial of a knot. On the other hand, this work illustrates the applicability and computational power of the employed computer algebra methods

Varias perspectivas sobre las bases de Gröbner : forma normal de Smith, algoritmo de Berlekamp y álgebras de Leibniz

En 1965 Buchberger introdujo el concepto de Base de Gröbner para un ideal del anillo de polinomios conmutativos y proporcionó un algoritmo de cálculo para calcular dichas bases. Desde entonces, la teoría de Bases de Gröbner ha experimentado un notable desarrollo, tanto en el terreno de sus aplicaciones que son abundantes y variadas, como en la extensión del concepto inicial de Base de Gröbner a otras estructuras matemáticas más complejas que el anillo de polinomios conmutativos. En este trabajo se incide en estas dos vertientes de las Bases de Gröbner.In 1965 Buchberger introduced the notion of Gröbner bases for a polynomial ideal and an algorithm for their computation. Since then, Gröbner bases has experimented a notable development in the area of its applications that they are abundant and assorted , as well as in the extension of the initial Gröbner bases concept to others more complex rings than the commutative polynomial ring

N-Koszul algebras, Calabi-Yau algebras and skew PBW extensions

Abstract. In the schematic approach to non-commutative algebraic geometry arises some important classes of non-commutative algebras like Koszul algebras, Artin-Schelter regular algebras, Calabi-Yau algebras, and closely related with them, the skew PBW extensions. There exist some relations between these algebras and the skew PBW extensions. We give conditions to guarantee that skew PBW extensions over fields are nonhomogeneous Koszul or Koszul algebras. We also show that a constant skew PBW extension of a field is a PBW deformation of its homogeneous version. We define graded skew PBW extensions, study some properties of these algebras and showed that if R is a PBW algebra then a graded skew PBW extension of R is a PBW algebra, and therefore, a Koszul algebra. As a generalization of the above results, we prove that every graded skew PBW extension of a finitely presented Koszul algebra is Koszul. Artin-Schelter regularity and the skew Calabi-Yau condition are studied for graded skew PBW extensions. We prove that every graded quasi-commutative skew PBW extension of an Artin-Schelter regular algebra is an Artin-Schelter regular algebra and, more general, graded skew PBW extensions of a finitely presented Auslander-regular algebra, are Artin-Schelter regular algebras. As a consequence, every graded quasi-commutative skew PBW extension of a finitely presented skew Calabi-Yau algebra is skew Calabi-Yau, and graded skew PBW extensions of a finitely presented Auslander-regular algebra are skew Calabi-Yau. Since graded quasi-commutative skew PBW extensions with coefficients in a finitely presented skew Calabi-Yau algebra are skew Calabi-Yau, the Nakayama automorphism exists for these extensions. With this in mind, we give a description of Nakayama automorphism for these non-commutative algebras using the Nakayama automorphism of the ring of the coefficients.En el enfoque esquemático de la geometría algebraica no conmutativa surgen algunas clases importantes de álgebras no conmutativas como álgebras de Koszul, álgebras Artin-Schelter regulares, álgebras Calabi-Yau y, estrechamente relacionadas con estas, las extensiones PBW torcidas. Existen algunas relaciones entre estas álgebras y las extensiones PBW torcidas. Nosotros damos condiciones para garantizar cuáles extensiones PBW torcidas de un cuerpo son álgebras no homogéneas de Koszul o álgebras de Koszul. También, mostramos que una extensión PBW torcida constante de un cuerpo es una deformación PBW de su versión homogénea. Definimos las extensiones PBW torcidas graduadas, estudiamos algunas propiedades de estas álgebras y mostramos que si R es un álgebra PBW, entonces cada extensión PBW torcida graduada de R es un álgebra PBW, y por lo tanto un álgebra de Koszul. Como una generalización de los resultados anteriores, se demuestra que cada extensión PBW torcida graduada de un álgebra de Koszul finitamente presentada, es un álgebra de Koszul. La regularidad de Artin-Schelter y la condición de Calabi-Yau torcida se estudian para las extensiones PBW torcidas graduadas. Se demuestra que cada extensión PBW torcida cuasi-conmutativa graduada de un álgebra Artin-Schelter regular es un álgebra Artin-Schelter regular, y más general, extensiones PBW torcidas graduadas de un álgebra finitamente presentada Auslander-regular, son álgebras Artin-Schelter regulares. Como consecuencia, cada extensión PBW torcida cuasi-conmutativa graduada de un álgebra Calabi-Yau torcida finitamente presentada, es Calabi-Yau torcida, y las extensiones PBW torcidas graduadas de un álgebra Auslander-regular finitamente presentada son álgebras Calabi-Yau torcidas. Dado que las extensiones PBW torcidas cuasi-conmutativas graduadas con coeficientes en un álgebra Calabi-Yau torcida finitamente presentada, son Calabi-Yau torcidas, existe el automorfismo de Nakayama para estas extensiones. Con esto en mente, damos una descripción del automorphism de Nakayama para estas álgebras no conmutativas, usando el automorphism de Nakayama del anillo de coeficientes.Doctorad

Fast Computation of the NN-th Term of a qq-Holonomic Sequence and Applications

33 pages. Long version of the conference paper Computing the $N$-th term of a $q$-holonomic sequence. Proceedings ISSAC'20, pp. 46–53, ACM Press, 2020 (https://hal.inria.fr/hal-02882885)International audienceIn 1977, Strassen invented a famous baby-step/giant-step algorithm that computes the factorial $N!$ in arithmetic complexity quasi-linear in $\sqrt{N}$. In 1988, the Chudnovsky brothers generalized Strassen’s algorithm to the computation of the $N$-th term of any holonomic sequence in essentially the same arithmetic complexity. We design $q$-analogues of these algorithms. We first extend Strassen’s algorithm to the computation of the $q$-factorial of $N$, then Chudnovskys' algorithm to the computation of the $N$-th term of any $q$-holonomic sequence. Both algorithms work in arithmetic complexity quasi-linear in $\sqrt{N}$; surprisingly, they are simpler than their analogues in the holonomic case. We provide a detailed cost analysis, in both arithmetic and bit complexity models. Moreover, we describe various algorithmic consequences, including the acceleration of polynomial and rational solving of linear $q$-differential equations, and the fast evaluation of large classes of polynomials, including a family recently considered by Nogneng and Schost