Bit-size estimates for triangular sets in positive dimension

Abstract

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 interpolation 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.

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

“…, X n ], and L U,U ′ (f 1 )(X 1 ) is given by Equation (20). From this equation and Equation (7), we deduce that lm( γ∈U 1 ℓ γ (X 1 )f 1 (γ)) = m, and a look at the explicit form of g i in Corollary 2 shows that lm(L U,U ′ (f 1 )(X 1 )) = X d 1 1 m, as expected.…”

“…The structure is also useful to express the polynomials in such Gröbner bases with interpolation formulas, in function of the solution points. Such formulas allow to get reasonably sharp upper bounds on the size of coefficients, as it was achieved for some special cases of lexicographic Gröbner bases in [7,8,6]. Another application is the possibility to decompose such "non-generic" Gröbner bases into smaller ones.…”

Irredundant Triangular Decomposition

“…• Consider algebraic sets defined by polynomials over Q and bound the height of the coefficients of the polynomials in the triangular sets. Such bit-size estimates were given for a single triangular set in the positive dimensional case in [9]. • Generalize Algorithm 3 to the case when the input system contains inequations.…”

“…Several authors, including Wu [27], Lazard [17], Kalkbrener [14], Wang [26], Moreno Maza [19], Schost [22], Chen [4,5], Dahan et al [8], have worked on triangular decompositions of algebraic sets, and some of these algorithms are implemented in M in the package RegularChains. ere exist sharp degree and height bounds for a single triangular set in terms of intrinsic data of the variety it represents, see for example the sequence of papers [21,22,7,9], these bounds are polynomial in the degree and the height of the variety. ere are also powerful randomized algorithms for computing triangular decompositions using Hensel li ing in the zero-dimensional case [8] and for irreducible varieties [21].…”

Heights of varieties in multiprojective spaces and arithmetic Nullstellensätze

“…The results in this direction are generically known as arithmetic Nullstellensätze and they play an important role in number theory and in theoretical computer science. In particular, they apply to problems in complexity and computability [Koi96,Asc04,DKS10], to counting problems over finite fields or over the rationals [BBK09,Rem10], and to effectivity in existence results in arithmetic geometry [KT08,BS10]. The first non-trivial result on this problem was obtained by Philippon, who got a bound on the minimal size of the denominator α in a Bézout identity as above [Phi90].…”