STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
A deterministic algorithm for sparse multivariate polynomial interpolation
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
On fast multiplication of polynomials over arbitrary algebras
Acta Informatica
Counting curves and their projections
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
Fast construction of irreducible polynomials over finite fields
Journal of Symbolic Computation
A polynomial time algorithm for diophantine equations in one variable
Journal of Symbolic Computation
Detecting perfect powers in essentially linear time
Mathematics of Computation
Modern computer algebra
Fast Probabilistic Algorithms for Verification of Polynomial Identities
Journal of the ACM (JACM)
Computing Jacobi symbols modulo sparse integers and polynomials and some applications
Journal of Algorithms
Probabilistic algorithms for sparse polynomials
EUROSAM '79 Proceedings of the International Symposiumon on Symbolic and Algebraic Computation
Mathematics of Computation
On square-free decomposition algorithms
SYMSAC '76 Proceedings of the third ACM symposium on Symbolic and algebraic computation
Early termination in sparse interpolation algorithms
Journal of Symbolic Computation - Special issue: International symposium on symbolic and algebraic computation (ISSAC 2002)
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
The matching problem for bipartite graphs with polynomially bounded permanents is in NC
SFCS '87 Proceedings of the 28th Annual Symposium on Foundations of Computer Science
On lacunary polynomial perfect powers
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Factoring Polynomials over Algebraic Number Fields
SIAM Journal on Computing
Supersparse black box rational function interpolation
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Factoring bivariate lacunary polynomials without heights
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
| Hi-index | 0.00 |
We consider solutions to the equation f=h^r for polynomials f and h and integer r=2. Given a polynomial f in the lacunary (also called sparse or super-sparse) representation, we first show how to determine if f can be written as h^r and, if so, to find such an r. This is a Monte Carlo randomized algorithm whose cost is polynomial in the number of non-zero terms of f and in logdegf, i.e., polynomial in the size of the lacunary representation, and it works over F"q[x] (for large characteristic) as well as Q[x]. We also give two deterministic algorithms to compute the perfect root h given f and r. The first is output-sensitive (based on the sparsity of h) and works only over Q[x]. A sparsity-sensitive Newton iteration forms the basis for the second approach to computing h, which is extremely efficient and works over both F"q[x] (for large characteristic) and Q[x], but depends on a number-theoretic conjecture. Work of Erdos, Schinzel, Zannier, and others suggests that both of these algorithms are unconditionally polynomial-time in the lacunary size of the input polynomial f. Finally, we demonstrate the efficiency of the randomized detection algorithm and the latter perfect root computation algorithm with an implementation in the C++ library NTL.