Factoring bivariate sparse (lacunary) polynomials

We present a deterministic algorithm for computing all irreducible factors of degree $\le d$ of a given bivariate polynomial $f\in K[x,y]$ 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 \Qbarra of degree $\le d$ which are not binomials can also be computed in time polynomial in the sparse length of the input and in $d$.Comment: 20 pp, Latex 2e. We learned on January 23th, 2006, that a multivariate version of Theorem 1 had independently been achieved by Erich Kaltofen and Pascal Koira

Lacunaryx: Computing bounded-degree factors of lacunary polynomials

In this paper, we report on an implementation in the free software Mathemagix of lacunary factorization algorithms, distributed as a library called Lacunaryx. These algorithms take as input a polynomial in sparse representation, that is as a list of nonzero monomials, and an integer $d$, and compute its irreducible degree-$\le d$ factors. The complexity of these algorithms is polynomial in the sparse size of the input polynomial and $d$.Comment: 6 page

Computing sparse multiples of polynomials

We consider the problem of finding a sparse multiple of a polynomial. Given f in F[x] of degree d over a field F, and a desired sparsity t, our goal is to determine if there exists a multiple h in F[x] of f such that h has at most t non-zero terms, and if so, to find such an h. When F=Q and t is constant, we give a polynomial-time algorithm in d and the size of coefficients in h. When F is a finite field, we show that the problem is at least as hard as determining the multiplicative order of elements in an extension field of F (a problem thought to have complexity similar to that of factoring integers), and this lower bound is tight when t=2.Comment: Extended abstract appears in Proc. ISAAC 2010, pp. 266-278, LNCS 650

A deterministic version of Pollard's p-1 algorithm

In this article we present applications of smooth numbers to the unconditional derandomization of some well-known integer factoring algorithms. We begin with Pollard's $p-1$ algorithm, which finds in random polynomial time the prime divisors $p$ of an integer $n$ such that $p-1$ is smooth. We show that these prime factors can be recovered in deterministic polynomial time. We further generalize this result to give a partial derandomization of the $k$-th cyclotomic method of factoring ($k\ge 2$) devised by Bach and Shallit. We also investigate reductions of factoring to computing Euler's totient function $\phi$. We point out some explicit sets of integers $n$ that are completely factorable in deterministic polynomial time given $\phi(n)$. These sets consist, roughly speaking, of products of primes $p$ satisfying, with the exception of at most two, certain conditions somewhat weaker than the smoothness of $p-1$. Finally, we prove that $O(\ln n)$ oracle queries for values of $\phi$ are sufficient to completely factor any integer $n$ in less than $\exp\Bigl((1+o(1))(\ln n)^{{1/3}} (\ln\ln n)^{{2/3}}\Bigr)$ deterministic time.Comment: Expanded and heavily revised version, to appear in Mathematics of Computation, 21 page

On Values of Cyclotomic Polynomials. V

In this paper, we present three results on cyclotomic polynomials. First, we present results about factorization of cyclotomic polynomials over arbitrary fields K. It is well known in cases such that a field K is the rational number field Q or a finite field F q (see [3, 4]). Using irreducibility of cyclotomic polynomials over Q, we can see that there are only finite elements of finite orders in a number field. On the other hand, we should correct some mistakes in [2, Corollary 1]. This mistake have no influence about another results in [2]. Finaly, we state about relations between Fibonacci polynomials and cyclotomic polynomials. This idea is due to K. Kuwano who stated this in his book [1] written in Japanese. 1. Factorizations of cyclotomic polynomials over fields The next theorem shows that irreducible factors of a cyclotomic polynomial Φn(x) over an arbi-trary field have the same degree. Theorem 1. Let K be a field. Then every irreducible factor f(x) of Φn(x) in K[x] has the same degree. More precisely, let L be the minimal splitting field of Φn(x) over a field K of characteristic p ≥ 0. Then we obtain that L is Galois over K, the Galois group G of L over K is a subgroup of the unit group of Z/mZ, where m = n in case p = 0 and n = pem with (m, p) = 1 in case p> 0, and deg f(x) = |G | = [L: K]. Proof. Let f(x) be a monic irreducible factor of Φn(x) in K[x] and let α ∈ L be a root of f(x). Then n = pem by [2, Theorem 1] where m is the order of α in L and m is not divided by p. Thus, we can see from the equation xm − 1 =∏d|m Φd(x) tha