Factoring sparse multivariate polynomials
Journal of Computer and System Sciences
Direct methods for sparse matrices
Direct methods for sparse matrices
Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
Factoring multivariate polynomials over algebraic number fields
SIAM Journal on Computing
Journal of Symbolic Computation - Special issue on computational algebraic complexity
On fast multiplication of polynomials over arbitrary algebras
Acta Informatica
Multivariate Polynomial Factorization
Journal of the ACM (JACM)
The Design and Analysis of Computer Algorithms
The Design and Analysis of Computer Algorithms
Factoring multivariate polynomials via partial differential equations
Mathematics of Computation
Hensel lifting and bivariate polynomial factorisation over finite fields
Mathematics of Computation
Factoring polynomials via polytopes
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
Complexity issues in bivariate polynomial factorization
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
Parallel methods for absolute irreducibility testing
The Journal of Supercomputing
Cache-oblivious polygon indecomposability testing
Proceedings of the 4th International Workshop on Parallel and Symbolic Computation
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A recent bivariate factorisation algorithm appeared in Abu-Salem et al. [Abu-Salem, F., Gao, S., Lauder, A., 2004. Factoring polynomials via polytopes. In: Proc. ISSAC'04. pp. 4-11] based on the use of Newton polytopes and a generalisation of Hensel lifting. Although possessing a worst-case exponential running time like the Hensel lifting algorithm, the polytope method should perform well for sparse polynomials whose Newton polytopes have very few Minkowski decompositions. A preliminary implementation in Abu-Salem et al. [Abu-Salem, F., Gao, S., Lauder, A., 2004. Factoring polynomials via polytopes. In: Proc. ISSAC'04. pp. 4-11] indeed reflects this property, but does not exploit the fact that the algorithm preserves the sparsity of the input polynomial, so that the total amount of work and space required are O(d^4) and O(d^2) respectively, for an input bivariate polynomial of total degree d. In this paper, we show that the polytope method can be made sensitive to the number of non-zero terms of the input polynomial, so that the input size becomes dependent on both the degree and the number of terms of the input bivariate polynomial. We describe a sparse adaptation of the polytope method over finite fields with prime order, which requires fewer bit operations and memory references given a degree d sparse polynomial whose number of terms t satisfies t