arXiv subjects
Publications and source records attributed to Jorge Jiménez Urroz.
Fix an integer $m$ bigger than 2. We prove that if there exists a finite group with $mp+1$ Suylow $p$-subgroups, where $p$ is large enough, then $mp+1$ is prime. This improves on a theorem of M. Hall and is a partial answer to Brauer's Problem 26. Our proof uses techniques from analytic number theory, and it also raises new questions in that area.
Every Latin square of prime power order $q$ is uniquely described by a local permutation polynomial (LPP) in the polynomial ring $\mathbb{F}_q[x,y]$. Despite this equivalence, one may find in the literature only some preliminary results on the relationship among Latin squares and LPPs. This paper delves into this topic by showing how the coefficients of any LPP are identified with the zeros of an algebraic set over $\mathbb{F}_q$. This allows for an algebraic description of all Latin squares of order $q$ by means of a unique polynomial in $\mathbb{F}_q[x,y]$, whose coefficients satisfy the constraints defined by the algebraic set under consideration. In order to make much easier the construction of this polynomial, we also deal with the natural translation to LPPs of both notions of reduced and isotopic Latin squares. Our algebraic approach is readily adapted to identify both types of Latin squares. All of the above is constructively illustrated for $q\in\{4,5\}$. We finish our study with the natural translation to LPPs of both notions of complete mappings and orthomorphisms of quasigroups, showing their relationship with transversals and isotopisms of Latin squares.
Maximally embedding dimension (MED) numerical semigroups are a wide and interesting family, with some remarkable algebraic and combinatorial properties. Associated to any numerical semigroup one can construct a MED closure, as it is well known. This paper shows two different explicit methods to construct this closure which also sheds new light on the very nature of this object.
Motivated by a model in quantum computation we study orthogonal sets of integral vectors of the same norm that can be extended with new vectors keeping the norm and the orthogonality. Our approach involves some arithmetic properties of the quaternions and other hypercomplex numbers.
We estimate the number of primes represented by a general quadratic polynomial with discriminant $Δ$, assuming that the corresponding real character is exceptional.