arXiv subjects

Xavier Dahan

Publications and source records attributed to Xavier Dahan.

9 recordsLinked to original sources

arXiv2021

On lexicographic Groebner bases of radical ideals in dimension zero: interpolation and structure

Due to the elimination property held by the lexicographic monomial order, the corresponding Groebner bases display strong structural properties from which meaningful informations can easily be extracted. We study these properties for radical ideals of (co)dimension zero. The proof presented relies on a combinatorial decomposition of the finite set of points whereby iterated Lagrange interpolation formulas permit to reconstruct a minimal Groebner basis. This is the first fully explicit interpolation formula for polynomials forming a lexicographic Groebner basis, from which the structure property can easily be read off. The inductive nature of the proof also yield as a byproduct a triangular decomposition algorithm from the Groebner basis.

cs.SC↗

arXiv2021

Lexicographic Groebner bases of bivariate polynomials modulo a univariate one

Let T(x) in k[x] be a monic non-constant polynomial and write R=k[x] / (T) the quotient ring. Consider two bivariate polynomials a(x, y), b(x, y) in R[y]. In a first part, T = p^e is assumed to be the power of an irreducible polynomial p. A new algorithm that computes a minimal lexicographic Groebner basis of the ideal ( a, b, p^e), is introduced. A second part extends this algorithm when T is general through the "local/global" principle realized by a generalization of "dynamic evaluation", restricted so far to a polynomial T that is squarefree. The algorithm produces splittings according to the case distinction "invertible/nilpotent", extending the usual "invertible/zero" in classic dynamic evaluation. This algorithm belongs to the Euclidean family, the core being a subresultant sequence of a and b modulo T. In particular no factorization or Groebner basis computations are necessary. The theoretical background relies on Lazard's structural theorem for lexicographic Groebner bases in two variables. An implementation is realized in Magma. Benchmarks show clearly the benefit, sometimes important, of this approach compared to the Groebner bases approach.

math.AC↗

arXiv2020

On a non-archimedean broyden method

Newton's method is an ubiquitous tool to solve equations, both in the archimedean and non-archimedean settings -- for which it does not really differ. Broyden was the instigator of what is called "quasi-Newton methods". These methods use an iteration step where one does not need to compute a complete Jacobian matrix nor its inverse. We provide an adaptation of Broyden's method in a general non-archimedean setting, compatible with the lack of inner product, and study its Q and R convergence. We prove that our adapted method converges at least Q-linearly and R-superlinearly with R-order $2^{\frac{1}{2m}}$ in dimension m. Numerical data are provided.

cs.SC↗

arXiv2020

The effects of coating culture dishes with collagen on fibroblast cell shape and swirling pattern formation

Motile human-skin fibroblasts form macroscopic swirling patterns when grown to confluence on a culture dish. In this paper, we investigate the effect of coating the culture-dish surface with collagen on the resulting pattern, using human-skin fibroblast NB1RGB cells as the model system. The presence of the collagen coating is expected to enhance the adherence of the fibroblasts to the dish surface, and thereby also enhance the traction that the fibroblasts have as they move. We find that, contrary to our initial expectation, the coating does not significantly affect the motility of the fibroblasts. Their eventual number density at confluence is also unaffected. However, the coherence length of cell orientation in the swirling pattern is diminished. We also find that the fibroblasts cultured in a collagen-coated dish are rounder in shape and shorter in perimeter, compared to those cultured in an uncoated dish. We hypothesize that the rounder cell-shape which weakens the cell-cell nematic contact interaction is responsible for the change in coherence length. A simple mathematical model of the migrating fibroblasts is constructed, which demonstrates that constant motility with weaker nematic interaction strength does indeed lead to the shortening of the coherence length.

physics.bio-ph↗

arXiv2018

Computation of gcd chain over the power of an irreducible polynomial

A notion of gcd chain has been introduced by the author at ISSAC 2017 for two univariate monic polynomials with coefficients in a ring R = k[x_1, ..., x_n ]/(T) where T is a primary triangular set of dimension zero. A complete algorithm to compute such a gcd chain remains challenging. This work treats completely the case of a triangular set T = (T_1 (x)) in one variable, namely a power of an irreducible polynomial. This seemingly "easy" case reveals the main steps necessary for treating the general case, and it allows to isolate the particular one step that does not directly extend and requires more care.

cs.SC↗

arXiv2017

On the bit-size of non-radical triangular sets

We present upper bounds on the bit-size of coefficients of non-radical lexicographical Groebner bases in purely triangular form (triangular sets) of dimension zero. This extends a previous work [Dahan-Schost, Issac'2004], constrained to radical triangular sets; it follows the same technical steps, based on interpolation. However, key notion of height of varieties is not available for points with multiplicities; therefore the bounds obtained are less universal and depend on some input data. We also introduce a related family of non- monic polynomials that have smaller coefficients, and smaller bounds. It is not obvious to compute them from the initial triangular set though.

cs.SC↗

arXiv2013

Regular graphs of large girth and arbitrary degree

For every integer d > 9, we construct infinite families {G_n}_n of d+1-regular graphs which have a large girth > log_d |G_n|, and for d large enough > 1,33 log_d |G_n|. These are Cayley graphs on PGL_2(q) for a special set of d+1 generators whose choice is related to the arithmetic of integral quaternions. These graphs are inspired by the Ramanujan graphs of Lubotzky-Philips-Sarnak and Margulis, with which they coincide when d is prime. When d is not equal to the power of an odd prime, this improves the previous construction of Imrich in 1984 where he obtained infinite families {I_n}_n of d+1-regular graphs, realized as Cayley graphs on SL_2(q), and which are displaying a girth > 0,48 log_d |I_n|. And when d is equal to a power of 2, this improves a construction by Morgenstern in 1994 where certain families {M_n}_n of 2^k+1-regular graphs were shown to have a girth > 2/3 log_d |M_n|.

math.CO↗

arXiv2012

Ramanujan graphs of very large girth based on octonions

We present a generalization of the construction of graphs by Lubotzky, Phillips and Sarnak in their celebrated article "Ramanujan graphs". The new approach consists in using octonion algebras rather than quaternions. A key tool is the existing result of the unique factorization of integral octonions. The families obtained by this mean present not only the same spectral property that make them good expanders, but also show a larger girth, yielding a new record for regular graphs.

math.CO↗

arXiv2010

Bit-size estimates for triangular sets in positive dimension

We give bit-size estimates for the coefficients appearing in triangular sets describing positive-dimensional algebraic sets defined over Q. These estimates are worst case upper bounds; they depend only on the degree and height of the underlying algebraic sets. We illustrate the use of these results in the context of a modular algorithm. This extends results by the first and last author, which were confined to the case of dimension 0. Our strategy is to get back to dimension 0 by evaluation and inter- polation techniques. Even though the main tool (height theory) remains the same, new difficulties arise to control the growth of the coefficients during the interpolation process.

cs.SC↗