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Factoring multivariate polynomials over finite fields. (English) Zbl 0577.12013
The author presents an algorithm for the factorization of multivariate polynomials with coefficients in a finite field which is polynomial-time in the degrees of the polynomial to be factored. This algorithm makes use of a new basis reduction algorithm for lattices over \({\mathbb{F}}_ q[Y].\)
In the case of two variables the algorithm is similar to the polynomial- time algorithm for the factorization of polynomials in one variable with rational coefficients [the author, H. W. Lenstra jun. and L. Lovász, Math. Ann. 261, 515-534 (1982; Zbl 0488.12001] and to that given by A. L. Chistov and D. Yu. Grigor’ev [Prepr. LOMI E-5- 82 (1982; Zbl 0509.68029)]. If \(f\in {\mathbb{F}}_ q[X_ 1,X_ 2,...,X_ t]\) for \(t>2\) the problem is reduced to the case \(t=2\) by substituting high enough powers of \(X_ 2\) for \(X_ 3\) up to \(X_ t\).

MSC:
11T06 Polynomials over finite fields
68W30 Symbolic computation and algebraic computation
12D05 Polynomials in real and complex fields: factorization
65Yxx Computer aspects of numerical algorithms
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