Linear Solving for Sign Determination

A general scheme in most algorithms dealing with the sign determination problem consists on the computation of the Tarski-query (also known as Sturm-query) for Z of many products of the given polynomials, in order to relate through a linear system this quantities with the number of elements in Z satisfying each possible sign condition. For instance, the naive approach were the Tarski-query of each of the 3 s polynomials of type P e 1 1 . . . P es s with e i = 0, 1, 2 for i = 1, . . . , s is computed, leads to a linear system of size 3 s × 3 s . Nevertheless, if m = #Z at most m sign conditions will be feasible, and then at most m of the coordinates of the solution of the considered linear system will be different from 0.

In [2], the exponential complexity arising from the number of Tarski-query computations and the resolution of the linear system in the approach described above is avoided. This is achieved by means of a recursive algorithm where the list P is divided in two sublists, the feasible sign conditions for each sublist is computed, and then this information is combined. Such combination is obtained by computing at most m 2 Tarski-queries and solving a linear system of size at most m 2 × m 2 .

In [6], [3] and [1, Chapter 10], the methods in [2] are further developed. In [6], an algorithm is given where the number of points in Z satisfying each feasible sign condition for the list P 1 , . . . , P i is computed sequentially for i = 1, . . . , s, following the idea that, at each step, each feasible sign condition for P 1 , . . . , P i-1 may be extended in at most 3 ways. To deal with the addition of the polynomial P i to the considered list, at most 2m Tarski-queries are computed and a linear system of size at most 3m × 3m is solved. In [3], a more explicit way to choose the polynomials whose Tarski-query is to be computed is given. In [1, Chapter 10], also the feasible sign conditions for P i on Z are computed at step i, in order to discard beforehand some non-feasible sign conditions for P 1 , . . . , P i on Z extending feasible sign conditions for P 1 , . . . , P i-1 on Z.

Depending on the setting, the Tarski-queries may be computed in different ways, taking a different number of operations in the field R, or in a proper domain D containing the coefficients of the polynomials in P and polynomials defining the finite set Z. As treated with general methods, the linear solving part can be done within O(m 2.376 ) operations in Q (see [4]). In [3,Section 3], where the univariate case is considered, the cost of linear solving dominates the overall complexity. In this work, we fix this situation by giving a specific method to solve with quadratic complexity the linear systems involved (Theorem 5). Even though this method can be used as a subroutine whenever these systems arise, we emphasize the result of its use in the univariate case. Following the complexity analisis in [3, Section 3.3] we obtain:

Corollary 1 Given P 0 , P 1 , . . . , P s ∈ R[X], P 0 ≡ 0, deg P i ≤ d for i = 0, . . . , s, the feasible sign conditions for P 1 , . . . , P s on {P 0 = 0} (and the number of elements in {P 0 = 0} satisfying each of these sign conditions) can be computed within O(sd

The motivation for this work also comes from probabilistic algorithms to determine feasible sign conditions in the multivariate case ( [5]), which produce a geometric resolution of a set of sample points. In this reduction to the univariate case, the degree of the polynomials obtained equals the Bézout number of some auxiliary polynomial systems, and the complexity of the algorithm depends quadratically on this quantity. Using Corollary 1, the treatment of the univariate case to find the feasible sign conditions for the original multivariate polynomials can be done without increasing the overall complexity.

We will follow mostly the notation in [1,Chapter 10]. In this reference the approach described in Section 1 is followed with the minor difference that the polynomials P 1 , . . . , P s are introduced one at each step from back to front; therefore, the notation is adapted to this order. For i = s, . . . , 1, we call P i the list P i , . . . , P s , and, at step i, we are given a list Σ = σ 1 , . . . , σ r of elements in {0, 1, -1} P i with σ 1 < lex • • • < lex σ r (0 ≺ 1 ≺ -1) containing, may be properly, all the feasible sign condition for P i on Z and we are to compute the exact list of feasible sign condition for P i on Z and the number of elements in Z satisfying each of these sign conditions. Note that the inequality r ≤ 3m holds at every step.

For P ∈ R[X], we note c(P = 0, Z), c(P > 0, Z) and c(P < 0, Z) the number of elements in Z satisfying the condition P = 0, P > 0 and P

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