Complexity of triangular representations of algebraic sets

Abstract

Triangular decomposition is one of the standard ways to represent the radical of a polynomial ideal. A general algorithm for computing such a decomposition was proposed by A. Szántó. In this paper, we give the first complete bounds for the degrees of the polynomials and the number of components in the output of the algorithm, providing explicit formulas for these bounds.

Irredundant Triangular Decomposition

“…(4) For every 1 i N and every ∈ ∆ i , the total degree of with respect to the free variables does not exceed (degW ) 2 and the degree with respect to every other variable does not exceed degW ; (5) All polynomials appearing in Algorithm 3 have total degrees bounded by…”

“…One of the main results of this paper is a new technique that ensures that triangular sets representing equidimensional components with different sets of free variables have no common irreducible components (see Step 2d of Algorithm 3 and Lemma 4.2). (2) e second difficulty is to factor out components that are embedded in higher dimensional irreducible components, similarly as it is stated in the Ri problem, mentioned above. is problem has only been solved for triangular sets in one and two dimensions [6,1].…”

“…Embedded components make it impossible to directly apply the intrinsic degree bounds. e best known degree bounds for the polynomials in a triangular decomposition are essentially D O (n 2 ) [10, 24,2] (D is a bound on the total degrees of f 1 , . .…”

A new bound on Hrushovski’s algorithm for computing the Galois group of a linear differential equation

“…In such a process, we need to take the differences between Gröbner bases and triangular sets into account. We apply Szántó's modified Wu-Ritt type decomposition algorithm [15,16] which has been proved to be more efficient than computing a Gröbner basis and make use of the numerical bound for Szántó's algorithm [1] to adapt to the complexity analysis of Hrushovski's algorithm. In doing this, we are able to avoid working with Gröbner bases to get a better bound of the degrees of the defining equations of the algebraic subgroups of F which is triple exponential in the order n of the given differential equation.…”

“…For example [9,Appendix], the differential Galois group of Bessel's equation t 2 y ′′ + ty ′ + (t 2 − ν 2 )y = 0 over C(t) is isomorphic to SL 2 (C) (not solvable) when ν ∈ 1 2 + Z. In other words, Bessel's equation cannot be solved by integrals, exponentials and algebraic functions unless ν ∈ 1 2 + Z. Computing the differential Galois groups would help us determine the existence of the solutions expressed in terms of elementary functions (integrals, exponentials and algebraic functions) and understand the algebraic relations among the solutions.…”