28 citations · 42 across the 9 of their papers we have counts for
9 papers
Lexicographic Groebner bases of bivariate polynomials modulo a univariate one
Xavier Dahan
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…
On a non-archimedean broyden method
Xavier Dahan, Tristan Vaccon
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…
Computation of gcd chain over the power of an irreducible polynomial
Xavier Dahan
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 primar…
On the bit-size of non-radical triangular sets
Xavier Dahan
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…
On lexicographic Groebner bases of radical ideals in dimension zero: interpolation and structure
Xavier Dahan
Due to the elimination property held by the lexicographic monomial order, the corresponding Groebner bases display strong structural properties from which meaningful informations c…
Structure of lexicographic Groebner bases in three variables of ideals of dimension zero
X. Dahan
We generalize the structural theorem of Lazard in 1985, from 2 variables to 3 variables. We use the Gianni-Kalkbrener result to do this, which implies some restrictions inside whic…
Regular graphs of large girth and arbitrary degree
Xavier Dahan
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 C…
Ramanujan graphs of very large girth based on octonions
Xavier Dahan, Jean-Pierre Tillich
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 octoni…
Bit-size estimates for triangular sets in positive dimension
Xavier Dahan, Abdulilah Kadri, Éric Schost
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 bou…