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Teresa Crespo

Publications and source records attributed to Teresa Crespo.

At least 19 recordsLinked to original sources
arXiv2025

Determining skew left braces of size np

We define the twofold semidirect product of two skew left braces, in which both the additive and multiplicative groups are semidirect products of the corresponding groups of the given skew left braces. We consider an odd prime $p$ and an integer $n$ satisfying $p\nmid n$, $p\nmid|\mathrm{Aut}(E)|$ for every group $E$ of order $n$ and such that each group of order $np$ has a unique $p$-Sylow subgroup. Under these conditions, we prove that any skew left brace of size $np$ is either a twofold semidirect product of the trivial brace of size $p$ and a skew left brace of size $n$ or a companion skew left brace of that one. We develop an algorithm to obtain all skew left braces of size $np$ from the skew left braces of size $n$ and provide a formula to count them. We use this result to describe all skew left braces of size $12p$ for $p\geq 7$, which proves a conjecture of V.G. Bardakov, M.V. Neshchadim and M.K. Yadav.

math.GR↗
arXiv2025

Left braces of size $p^2q^2$

We consider relatively prime integer numbers $m$ and $n$ such that each solvable group of order $mn$ has a normal subgroup of order $m$. We prove that each brace of size $mn$ is a semidirect product of a brace of size $m$ and a brace of size $n$. We further give a method to classify braces of size $mn$ from the classification of braces of sizes $m$ and $n$. We apply this result to determine all braces of size $p^2q^2$, for $p$ and $q$ odd primes satisfying some conditions which hold in particular for $p$ a Germain prime and $q=2p+1$.

math.GR↗
arXiv2022

Left braces of size 8p

We describe all left braces of size 8p for p an odd prime different from 3 or 7 and validate the number given by Bardakov, Neschadim and Yadav. We give a characterization for isomorphism classes of a semidirect product of left braces and then the description is done by first describing left braces of size 8, as conjugacy classes of regular subgroups of the corresponding holomorph, and then checking how many non isomorphic left braces of size 8p are obtained from each one of them.

math.GR↗
arXiv2022

Inducing braces and Hopf Galois structures

Let $p$ be a prime number and let $n$ be an integer not divisible by $p$ and such that every group of order $np$ has a normal subgroup of order $p$. (This holds in particular for $p>n$.) We prove that left braces of size $np$ may be obtained as a semidirect product of the unique left brace of size $p$ and a left brace of size $n$. We give a method to determine all braces of size $np$ from the braces of size $n$ and certain classes of morphisms from the multiplicative group of these braces of size $n$ to $\mathrm{Z}_p^*$. From it we derive a formula giving the number of Hopf Galois structures of abelian type $\mathrm{Z}_p \times E$ on a Galois extension of degree $np$ in terms of the number of Hopf Galois structures of abelian type $E$ on a Galois extension of degree $n$. For a prime number $p\geq 7$, we apply the obtained results to describe all left braces of size $12p$ and determine the number of Hopf Galois structures of abelian type on a Galois extension of degree $12p$.

math.NT↗
arXiv2021

Real Liouvillian Extensions of Partial Differential Fields

In this paper, we establish Galois theory for partial differential systems defined over formally real differential fields with a real closed field of constants and over formally $p$-adic differential fields with a $p$-adically closed field of constants. For an integrable partial differential system defined over such a field, we prove that there exists a formally real (resp. formally $p$-adic) Picard-Vessiot extension. Moreover, we obtain a uniqueness result for this Picard-Vessiot extension. We give an adequate definition of the Galois differential group and obtain a Galois fundamental theorem in this setting. We apply the obtained Galois correspondence to characterise formally real Liouvillian extensions of real partial differential fields with a real closed field of constants by means of split solvable linear algebraic groups. We present some examples of real dynamical systems and indicate some possibilities of further development of algebraic methods in real dynamical systems.

math.RA↗
arXiv2021

Picard-Vessiot theory for real partial differential fields

We prove the existence of real Picard-Vessiot extensions for real partial differential fields with real closed field of constants. We establish a Galois correspondence theorem for these Picard-Vessiot extensions and characterize real Liouville extensions of real partial differential fields.

math.AG↗
arXiv2020

Automatic realization of Hopf Galois structures

We consider Hopf Galois structures on a separable field extension $L/K$ of degree $p^n$, for $p$ an odd prime number, $n\geq 3$. For $p > n$, we prove that $L/K$ has at most one abelian type of Hopf Galois structures. For a nonabelian group $N$ of order $p^n$, with commutator subgroup of order $p$, we prove that if $L/K$ has a Hopf Galois structure of type $N$, then it has a Hopf Galois structure of type $A$, where $A$ is an abelian group of order $p^n$ and having the same number of elements of order $p^m$ as $N$, for $1\leq m \leq n$.

math.GR↗
arXiv2019

Picard-Vessiot Extensions of Real Differential Fields

For a linear differential equation defined over a formally real differential field K with real closed field of constants k, Crespo, Hajto and van der Put proved that there exists a unique formally real Picard- Vessiot extension up to K-differential automorphism. However such an equation may have Picard-Vessiot extensions which are not formally real fields. The differential Galois group of a Picard-Vessiot extension for this equation has the structure of a linear algebraic group defined over k and is a k-form of the differential Galois group H of the equation over the differential field K(i), where i denotes a square root of -1 in the algebraic closure of k. These facts lead us to consider two issues: determining the number of K-differential isomorphism classes of Picard-Vessiot extensions and describing the variation of the differential Galois group in the set of k-forms of H. We address these two issues in the cases when H is a special linear, a special orthogonal, or a symplectic linear algebraic group and conclude that there is no general behaviour.

math.AG↗
arXiv2019

Jacobian Conjecture via Differential Galois Theory

We prove that a polynomial map is invertible if and only if some associated differential ring homomorphism is bijective. To this end, we use a theorem of Crespo and Hajto linking the invertibility of polynomial maps with Picard-Vessiot extensions of partial differential fields, the theory of strongly normal extensions as presented by Kovacic and the characterization of Picard-Vessiot extensions in terms of tensor products given by Levelt.

math.AG↗
arXiv2019

Computation of Hopf Galois structures on separable extensions and classification of those for degree twice an odd prime power

A Hopf Galois structure on a finite field extension $L/K$ is a pair $(H,μ)$, where $H$ is a finite cocommutative $K$-Hopf algebra and $μ$ a Hopf action. In this paper we present a program written in the computational algebra system Magma which gives all Hopf Galois structures on separable field extensions of a given degree and several properties of those. We show a table which summarizes the program results. Besides, for separable field extensions of degree $2p^n$, with $p$ an odd prime number, we prove that the occurrence of some type of Hopf Galois structure may either imply or exclude the occurrence of some other type. In particular, for separable field extensions of degree $2p^2$, we determine exactly the possible sets of Hopf Galois structure types.

math.GR↗
arXiv2018

Hopf Galois structures on separable field extensions of odd prime power degree

A Hopf Galois structure on a finite field extension $L/K$ is a pair $(\mathcal{H},μ)$, where $\mathcal{H}$ is a finite cocommutative $K$-Hopf algebra and $μ$ a Hopf action. In this paper, we present several results on Hopf Galois structures on odd prime power degree separable field extensions. We prove that if a separable field extension of odd prime power degree has a Hopf Galois structure of cyclic type, then it has no structure of noncyclic type. We determine the number of Hopf Galois structures of cyclic type on a separable field extension of degree $p^n$, $p$ an odd prime, such that the Galois group of its normal closure is a semidirect product $C_{p^n}\rtimes C_D$ of the cyclic group of order $p^n$ and a cyclic group of order $D$, with $D$ prime to $p$. We characterize the transitive groups of degree $p^3$ which are Galois groups of the normal closure of a separable field extension having some cyclic Hopf Galois structure and determine the number of those. We prove that if a separable field extension of degree $p^3$ has a nonabelian Hopf Galois structure then it has an abelian structure whose type has the same exponent as the nonabelian type. We obtain that, for $p>3$, the two abelian noncyclic Hopf Galois structures do not occur on the same separable extension of degree $p^3$. We present a table which gives the number of Hopf Galois structures of each possible type on a separable extension of degree $27$ to illustrate that for $p=3$, all four noncyclic Hopf Galois structures may occur on the same extension. Finally, putting together all previous results, we list all possible sets of Hopf Galois structure types on a separable extension of degree $p^3$, for $p>3$ a prime.

math.GR↗
arXiv2018

Computation of Hopf Galois structures on low degree separable extensions and classification of those for degrees $p^2$ and $2p$

A Hopf Galois structure on a finite field extension $L/K$ is a pair $(H,μ)$, where $H$ is a finite cocommutative $K$-Hopf algebra and $μ$ a Hopf action. In this paper we present a program written in the computational algebra system Magma which gives all Hopf Galois structures on separable field extensions of degree up to eleven and several properties of those. Besides, we exhibit several results on Hopf Galois structures inspired by the program output. We prove that if $(H,μ)$ is an almost classically Hopf Galois structure, then it is the unique Hopf Galois structure with underlying Hopf algebra $H$, up to isomorphism. For $p$ an odd prime, we prove that a separable extension of degree $p^2$ may have only one type of Hopf Galois structure and determine those of cyclic type; we determine as well the Hopf Galois structures on separable extensions of degree $2p$. We highlight the richness of the results obtained for extensions of degree 8 by computing an explicit example and presenting some tables which summarizes these results.

math.GR↗
arXiv2018

Hopf Galois structures on symmetric and alternating extensions

By using our previous results on induced Hopf Galois structures and a recent result by Koch, Kohl, Truman and Underwood on normality, we determine which types of Hopf Galois structures occur on Galois extensions with Galois group isomorphic to alternating or symmetric groups.

math.GR↗
arXiv2017

An algorithm to determine Hopf Galois structures

A Hopf Galois structure on a finite field extension L/K is given by a finite cocommutative K-Hopf algebra and a Hopf action. In this paper we present an algorithm written in the computational algebra system Magma which gives all Hopf Galois structures on separable field extensions of a given degree and several properties of those. We describe the results obtained for extensions of degree up to 11. Besides, we prove that separable extensions of degree equal to the square of an odd prime have at most one type of Hopf Galois structures.

math.GR↗
arXiv2016

On some nice polynomial automorphisms

Given a polynomial endomorphism F of the n-dimensional affine space over a field K, we define a sequence of polynomial endomorphisms of the affine space associated to F. We call F nice if there exists an integer m such that the m-th term of the sequence vanishes. Then F is invertible and its inverse may be computed in terms of the endomorphisms in the sequence. In this paper we study the class of nice polynomial automorphisms and obtain that this class is large and includes triangulable automorphisms and all linear cubic homogeneous polynomial automorphisms of nilpotence index up to 3. We prove as well that the nicety property is invariant under linear conjugation and determine that seven of the eight forms in Hubbers' classification of cubic homogeneous automorphisms in dimension 4 are nice and moreover the eighth one is a composition of nice maps.

math.AG↗
arXiv2016

A new characterization of the invertibility of polynomial maps

In this paper we present an equivalent statement to the Jacobian conjecture. For a polynomial map F on an affine space of dimension n, we define recursively n finite sequences of polynomials. We give an equivalent condition to the invertibility of F as well as a formula for the inverse of F in terms of these finite sequences of polynomials. Some examples illustrate the effective aspects of our approach.

math.AC↗
arXiv2015

Jacobian Conjecture and Nilpotency

For K a field of characteristic 0 and d any integer number greater than or equal to 2, we prove the invertibility of polynomial endomorphisms of the affine space of dimension d over K of the form F=Id+H, where each coordinate of H is the cube of a linear form and the cube of the Jacobian matrix of H is equal to zero. Our proof uses the inversion algorithm for polynomial maps presented in our previous paper. Our current result leads us to formulate a conjecture relating the nilpotency degree of the Jacobian matrix of H with the number of necessary steps in the inversion algorithm.

math.AG↗