Irreducibility of multivariate polynomials
Journal of Computer and System Sciences
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
Subresultants and Reduced Polynomial Remainder Sequences
Journal of the ACM (JACM)
On Euclid's Algorithm and the Computation of Polynomial Greatest Common Divisors
Journal of the ACM (JACM)
Fast Probabilistic Algorithms for Verification of Polynomial Identities
Journal of the ACM (JACM)
Probabilistic algorithms for sparse polynomials
EUROSAM '79 Proceedings of the International Symposiumon on Symbolic and Algebraic Computation
STOC '73 Proceedings of the fifth annual ACM symposium on Theory of computing
ACM '73 Proceedings of the ACM annual conference
Uniform closure properties of P-computable functions
STOC '86 Proceedings of the eighteenth annual ACM symposium on Theory of computing
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
A system for manipulating polynomials given by straight-line programs
SYMSAC '86 Proceedings of the fifth ACM symposium on Symbolic and algebraic computation
Greatest common divisors of polynomials given by straight-line programs
Journal of the ACM (JACM)
Interpolating polynomials from their values
Journal of Symbolic Computation - Special issue on computational algebraic complexity
On the multi-threaded computation of integral polynomial greatest common divisors
ISSAC '91 Proceedings of the 1991 international symposium on Symbolic and algebraic computation
Interpolation of polynomials given by straight-line programs
Theoretical Computer Science
Hierarchical representations with signatures for large expression management
AISC'06 Proceedings of the 8th international conference on Artificial Intelligence and Symbolic Computation
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We develop algorithms on multivariate polynomials represented by straight-line programs for the greatest common divisor problem and conversion to sparse representation. Our algorithms are in random polynomial-time for the usual coefficient fields and output with controllably high probability the correct result which for the GCD problem is a straight-line program determining the GCD of the inputs and for the conversion algorithm is the sparse representation of the input. The algorithms only require an a priori bound for the total degrees of the inputs. Over rational numbers the conversion algorithm also needs a bound on the size of the polynomial coefficients. As specializations we get, e.g., random polynomial-time algorithms for computing the sparse GCD of polynomial determinants or for computing the sparse solution of a linear system whose coefficients are given by formulas.