The null space problem I. complexity
SIAM Journal on Algebraic and Discrete Methods
Greatest common divisors of polynomials given by straight-line programs
Journal of the ACM (JACM)
Journal of Symbolic Computation - Special issue on computational algebraic complexity
A polynomial time algorithm for diophantine equations in one variable
Journal of Symbolic Computation
Cryptanalysis of the HFE Public Key Cryptosystem by Relinearization
CRYPTO '99 Proceedings of the 19th Annual International Cryptology Conference on Advances in Cryptology
Early termination in sparse interpolation algorithms
Journal of Symbolic Computation - Special issue: International symposium on symbolic and algebraic computation (ISSAC 2002)
Algorithms for computing sparsest shifts of polynomials in power, Chebyshev, and Pochhammer bases
Journal of Symbolic Computation - Special issue: International symposium on symbolic and algebraic computation (ISSAC 2002)
On the complexity of computing determinants
Computational Complexity
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
On probabilistic analysis of randomization in hybrid symbolic-numeric algorithms
Proceedings of the 2007 international workshop on Symbolic-numeric computation
On exact and approximate interpolation of sparse rational functions
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
On lacunary polynomial perfect powers
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Interpolation of polynomials given by straight-line programs
Theoretical Computer Science
Proceedings of the 4th International Workshop on Parallel and Symbolic Computation
Interpolation of Shifted-Lacunary Polynomials
Computational Complexity
Detecting lacunary perfect powers and computing their roots
Journal of Symbolic Computation
Decoding by linear programming
IEEE Transactions on Information Theory
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
Sparse multivariate function recovery from values with noise and outlier errors
Proceedings of the 38th international symposium on 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
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We present a method for interpolating a supersparse blackbox rational function with rational coefficients, for example, a ratio of binomials or trinomials with very high degree. We input a blackbox rational function, as well as an upper bound on the number of non-zero terms and an upper bound on the degree. The result is found by interpolating the rational function modulo a small prime p, and then applying an effective version of Dirichlet's Theorem on primes in an arithmetic progression progressively lift the result to larger primes. Eventually we reach a prime number that is larger than the inputted degree bound and we can recover the original function exactly. In a variant, the initial prime p is large, but the exponents of the terms are known modulo larger and larger factors of p-1. The algorithm, as presented, is conjectured to be polylogarithmic in the degree, but exponential in the number of terms. Therefore, it is very effective for rational functions with a small number of non-zero terms, such as the ratio of binomials, but it quickly becomes ineffective for a high number of terms. The algorithm is oblivious to whether the numerator and denominator have a common factor. The algorithm will recover the sparse form of the rational function, rather than the reduced form, which could be dense. We have experimentally tested the algorithm in the case of under 10 terms in numerator and denominator combined and observed its conjectured high efficiency.