Sharp estimates for triangular sets

Abstract

We study the triangular representation of zero-dimensional varieties defined over the rational field (resp. a rational function field). We prove polynomial bounds in terms of intrinsic quantities for the height (resp. degree) of the coefficients of such triangular sets, whereas previous bounds were exponential. We also introduce a rational form of triangular representation, for which our estimates become linear. Experiments show the practical interest of this new representation.

Sparse Rational Univariate Representation

“…To this end we have restated known results for the parametrization of the roots, see for example [1,15]. However, we use resultant computations and especially the Poisson formula for the resultant to express the various quantities.…”

“…The existing estimates on the degree and the height of the polynomial involved in the representation are based on total degree or ISSAC '17, July [25][26][27][28]2017 Bézout bounds, on the height theory of varieties and on the theory of Chow forms, [1,15,35]. This representation is related to the arithmetic Nullstellensätz [18,30,39,40], see also [33] for the most recent approach, and the separation bounds of the polynomial systems [21].…”

A Modular Method for Computing the Splitting Field of a Polynomial

“…Thus this reduced Gröbner basis can be obtained by "algebraic factoring methods" (see [2]) and is said to be a triangular basis (see [11,6]). For a Gröbner basis G ⊂ Q[X] and a polynomial P , let NF(P, G) denote the normal form of P in Q[X] with respect to G (see [5]).…”

“…We emphasise that one can combine other methods for the computation of G with the proposed scheme. For example we could combine sparse interpolations strategy effectively (dense interpolation formulas are given in [6,12]), this will be study in a future work. We also note that it is possible to translate the results presented in this article to polynomials over global fields.…”

“…the integration of sparse interpolation formulas (like the dense ones in [6,12]) in our algorithm. Also, it would be interesting to study the possibility of using this approach in a dynamical strategy like the one of Magma (see [19]).…”