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10
Factoring bivariate sparse (lacunary) polynomials
 J. Complexity
"... Abstract. We present a deterministic algorithm for computing all irreducible factors of degree ≤ d of a given bivariate polynomial f ∈ K[x,y] over an algebraic number field K and their multiplicities, whose running time is polynomial in the bit length of the sparse encoding of the input and in d. Mo ..."
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Abstract. We present a deterministic algorithm for computing all irreducible factors of degree ≤ d of a given bivariate polynomial f ∈ K[x,y] over an algebraic number field K and their multiplicities, whose running time is polynomial in the bit length of the sparse encoding of the input and in d. Moreover, we show that the factors over Q of degree ≤ d which are not binomials can also be computed in time polynomial in the sparse length of the input and in d.
The Number of Real Roots of a Bivariate Polynomial on a Line,” submitted for publication
"... We prove that a polynomial f ∈ R[x, y] with t nonzero terms, restricted on a real line y = ax+b, either has at most 6t−4 zeroes or vanishes over the whole line. As a consequence, we derive an alternative algorithm to decide whether a linear polynomial y−ax−b ∈ K[x, y] divides a sparse polynomial f ..."
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We prove that a polynomial f ∈ R[x, y] with t nonzero terms, restricted on a real line y = ax+b, either has at most 6t−4 zeroes or vanishes over the whole line. As a consequence, we derive an alternative algorithm to decide whether a linear polynomial y−ax−b ∈ K[x, y] divides a sparse polynomial f ∈ K[x, y] with t terms in [log(H(f)H(a)H(b))[K: Q] log(deg(f))t] O(1) bit operations, where K is a real number field. 1
Detecting lacunary perfect powers and computing their roots
, 2009
"... We consider the problem of determining whether a lacunary (also called a sparse or supersparse) polynomial f is a perfect power, that is, f = h r for some other polynomial h and r ∈ N, and of finding h and r should they exist. We show how to determine if f is a perfect power in time polynomial in t ..."
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We consider the problem of determining whether a lacunary (also called a sparse or supersparse) polynomial f is a perfect power, that is, f = h r for some other polynomial h and r ∈ N, and of finding h and r should they exist. We show how to determine if f is a perfect power in time polynomial in the number of nonzero terms of f, and in terms of log deg f, i.e., polynomial in the size of the lacunary representation. The algorithm works over Fq[x] (for large characteristic) and over Z[x], where the cost is also polynomial in log ‖f‖∞. We also give a Monte Carlo algorithm to find h if it exists, for which our proposed algorithm requires polynomial time in the output size, i.e., the sparsity and height of h. Conjectures of Erdös and Schinzel, and recent work of Zannier, suggest that h must be sparse. Subject to a slightly stronger conjectures we give an extremely efficient algorithm to find h via a form of sparse Newton iteration. We demonstrate the efficiency of these algorithms with an implementation using the C++ library NTL. 1.
Reduction of bivariate polynomials from convexdense to dense, with application to factorizations
 Math. Comp
"... Abstract. In this article we present a new algorithm for reducing the usual sparse bivariate factorization problems to the dense case. This reduction simply consists of computing an invertible monomial transformation that produces a polynomial with a dense size of the same order of magnitude as the ..."
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Abstract. In this article we present a new algorithm for reducing the usual sparse bivariate factorization problems to the dense case. This reduction simply consists of computing an invertible monomial transformation that produces a polynomial with a dense size of the same order of magnitude as the size of the integral convex hull of the support of the input polynomial. This approach turns out to be very efficient in practice, as demonstrated with our implementation. 1.
FEWNOMIAL SYSTEMS WITH MANY ROOTS, AND AN ADELIC TAU CONJECTURE
"... Abstract. Consider a system F of n polynomials in n variables, with a total of n+k distinct exponent vectors, over any local field L. We discuss conjecturally tight bounds on the maximal number of nondegenerate roots F can have over L, with all coordinates having fixed phase, as a function of n, k, ..."
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Abstract. Consider a system F of n polynomials in n variables, with a total of n+k distinct exponent vectors, over any local field L. We discuss conjecturally tight bounds on the maximal number of nondegenerate roots F can have over L, with all coordinates having fixed phase, as a function of n, k, and L only. In particular, we give new explicit systems with number of roots approaching the best known upper bounds. We also briefly review the background behind such bounds, and their application, including connections to computational number theory and variants of the ShubSmale τConjecture and the P vs. NP Problem. One of our key tools is the construction of combinatorially constrained tropical varieties with maximally many intersections.
Supersparse black box rational function interpolation
 Manuscript
, 2011
"... 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 nonzero terms and an upper bound on the ..."
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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 nonzero 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 nonzero 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.
Optimizing nvariate (n+k)nomials for small k
, 2010
"... We give a high precision polynomialtime approximation scheme for the supremum of any honest nvariate (n + 2)nomial with a constant term, allowing real exponents as well as real coefficients. Our complexity bounds count field operations and inequality checks, and are quadratic in n and the logarit ..."
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We give a high precision polynomialtime approximation scheme for the supremum of any honest nvariate (n + 2)nomial with a constant term, allowing real exponents as well as real coefficients. Our complexity bounds count field operations and inequality checks, and are quadratic in n and the logarithm of a certain condition number. For the special case of nvariate (n+2)nomials with integer exponents, the log of our condition number is subquadratic in the sparse size. The best previous complexity bounds were exponential in the sparse size, even for n fixed. Along the way, we partially extend the theory of Viro diagrams and Adiscriminants to real exponents. We also show that, for any fixed δ>0, deciding whether the supremum of an nvariate ( n+n δ)nomial exceeds a given number is NPRcomplete.
A Hitting Set Construction, with Applications to Arithmetic Circuit Lower Bounds
, 2009
"... Abstract. A polynomial identity testing algorithm must determine whether a given input polynomial is identically equal to 0. We give a deterministic blackbox identity testing algorithm for univariate polynomials of the form Pt j=0 cjXα j β (a + bX) j. From our algorithm we derive Q an exponential l ..."
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Abstract. A polynomial identity testing algorithm must determine whether a given input polynomial is identically equal to 0. We give a deterministic blackbox identity testing algorithm for univariate polynomials of the form Pt j=0 cjXα j β (a + bX) j. From our algorithm we derive Q an exponential lower bound for representations of polynomials such as n 2 i=1 (Xi − 1) under this form. It has been conjectured that these polynomials are hard to compute by general arithmetic circuits. Our result shows that the “hardness from derandomization” approach to lower bounds is feasible for a restricted class of arithmetic circuits. The proof is based on techniques from algebraic number theory, and more precisely on properties of the height function of algebraic numbers. 1
Optimizing nvariate (n + k)nomials for small k
, 2010
"... We give a high precision polynomialtime approximation scheme for the supremum of any honest nvariate (n+2)nomial with a constant term, allowing real exponents as well as real coefficients. Our complexity bounds count field operations and inequality checks, and are polynomial in n and the logarith ..."
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We give a high precision polynomialtime approximation scheme for the supremum of any honest nvariate (n+2)nomial with a constant term, allowing real exponents as well as real coefficients. Our complexity bounds count field operations and inequality checks, and are polynomial in n and the logarithm of a certain condition number. For the special case of polynomials (i.e., integer exponents), the log of our condition number is subquadratic in the sparse size. The best previous complexity bounds were exponential in the sparse size, even for n fixed. Along the way, we partially extend the theory of Adiscriminants to real exponents and exponential sums, and find new and natural NPRcomplete problems.