We study bounds for the Castelnuovo–Mumford regularity of homogeneous ideals in a polynomial ring in terms of the number of variables and the degree of the generators. In particular, our aim is to give a positive answer to a question posed by Bayer and Mumford in What can be computed in algebraic geometry? (Computational algebraic geometry and commutative algebra, Symposia Mathematica, vol. XXX...

This article comes from our quest for bounds on the Castelnuovo-Mumford regularity of schemes in terms of their “defining equations”, in the spirit of [BM], [BEL], [CP] or [CU]. The references [BS], [BM], [V] or [C] explains how this notion of regularity is a mesure of the algebraic complexity of the scheme, and provides several computational motivations. It was already remarked by several auth...

The Eisenbud–Goto conjecture states that reg X ≤ deg − codim + 1 for a nondegenerate irreducible projective variety over an algebraically closed field. While this is known to be false in general, it has been proven several special cases, including when toric of codimension 2. We classify the varieties 2 having maximal regularity, is, which equality holds bound. also give combinatorial character...

We give an alternative proof of the regularity, up to loose end, minimizers, resp. critical points Mumford-Shah functional when they are sufficiently close cracktip, consist a single arc terminating at interior point. On donne une nouvelle preuve de régularité, jusqu'a l'extrémité libre, des minimums (respectivement, critiques) la fonctionnelle qui sont suffisamment proche du cracktip ils compo...

We improve certain degree bounds for Gröbner bases of polynomial ideals in generic position. We work exclusively in deterministically verifiable and achievable generic positions of a combinatorial nature, namely either strongly stable position or quasi stable position. Furthermore, we exhibit new dimension(and depth-)dependent upper bounds for the Castelnuovo-Mumford regularity and the degrees ...

We prove two recent conjectures on some upper bounds for the Castelnuovo-Mumford regularity of the binomial edge ideals of some different classes of graphs. We prove the conjecture of Matsuda and Murai for chordal graphs. We also prove the conjecture due to the authors for a class of chordal graphs. We determine the regularity of the binomial edge ideal of the join of graphs in terms of the reg...