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Communication Dans Un Congrès Année : 2021

On exact division and divisibility testing for sparse polynomials

Pascal Giorgi
Bruno Grenet
Armelle Perret Du Cray
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Résumé

No polynomial-time algorithm is known to test whether a sparse polynomial G divides another sparse polynomial F . While computing the quotient Q = F quo G can be done in polynomial time with respect to the sparsities of F , G and Q, this is not yet sufficient to get a polynomial-time divisibility test in general. Indeed, the sparsity of the quotient Q can be exponentially larger than the ones of F and G. In the favorable case where the sparsity #Q of the quotient is polynomial, the best known algorithm to compute Q has a non-linear factor #G#Q in the complexity, which is not optimal. In this work, we are interested in the two aspects of this problem. First, we propose a new randomized algorithm that computes the quotient of two sparse polynomials when the division is exact. Its complexity is quasi-linear in the sparsities of F , G and Q. Our approach relies on sparse interpolation and it works over any finite field or the ring of integers. Then, as a step toward faster divisibility testing, we provide a new polynomial-time algorithm when the divisor has a specific shape. More precisely, we reduce the problem to finding a polynomial S such that QS is sparse and testing divisibility by S can be done in polynomial time. We identify some structure patterns in the divisor G for which we can efficiently compute such a polynomial S.

Dates et versions

hal-03136945 , version 1 (10-02-2021)

Identifiants

Citer

Pascal Giorgi, Bruno Grenet, Armelle Perret Du Cray. On exact division and divisibility testing for sparse polynomials. ISSAC 2021 - 46th International Symposium on Symbolic and Algebraic Computation, Jul 2021, Saint-Pétersbourg, Russia. pp.163-170, ⟨10.1145/3452143.3465539⟩. ⟨hal-03136945⟩
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