In this paper we present a polynomial-time algorithm to solve the following problem: given a non-zero polynomial fe Q[X] in one variable with rational coefficients, find the decomposition of f into irreducible factors in Q[X]. It is well known that this is equivalent to factoring primitive polynomials fe [X] into irreducible factors in 7L[X]. Here we call fe7L[X] primitive if the greatest common divisor of its coefficients (the content of f) is 1. Our algorithm performs well in practice, cf. [8]. Its running time, measured in bit operations, is O(n 12 +n 9 (logfI) 3). Here fE[X] is the polynomial to be factored, n = deg(f) is the degree of J and ZaiXi = for a polynomial Za 1 X 1 with real coefficients a 1. An outline of the algorithm is as follows. First we find, for a suitable small prime number p, a p-adic irreducible factor h of J to a certain precision. This is done with Berlekamp's algorithm for factoring polynomials over small finite fields, combined with Hensel's lemma. Next we look for the irreducible factor h 0 of f in ZL[X] that is divisible by h. The condition that h 0 is divisible by h means that h 0 belongs to a certain lattice, and the condition that h 0 divides f implies that the coefficients of h 0 are relatively small. It follows that we must look for a "small" element in that lattice, and this is done by means of a basis reduction algorithm. It turns out that this enables us to determine h 0. The algorithm is repeated until all irreducible factors of f have been found. The basis reduction algorithm that we employ is new, and it is described and analysed in Sect. 1. It improves the algorithm given in a preliminary version of [9, Sect. 3]. At the end of Sect. 1 we briefly mention two applications of the new algorithm to diophantine approximation. The connection between factors off and reduced bases of a lattice is treated in detail in Sect. 2. The theory presented here extends a result appearing in [8, Theorem 2]. It should be remarked that the latter result, which is simpler to prove, would in principle have sufficed for our purpose.

It is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable. The proof depends on methods from geometry of numbers.

The motivation for this work came from the desire to understand heuristically (since proofs seem out of reach at present) a number of experimental observations about class groups of number fields, and in particular imaginary and real quadratic fields. In turn the heuristic explanations that we obtain may help to find the way towards a proof.Three of these observations are äs follows :A_/ The odd part of the class group of an imaginary quadratic field seems to be quite rarely non cyclic.B/ If P is a small odd prime, the proportion of imaginary quadratic fields whose class number is divisible by p seems to be significantly greater than l/p (for instance 43% for p = 3, 23.5% for p=5).C/ It seems that a definite non zero proportion of real quadratic fields of prime discriirinant (close to 76%) has class number l , although it is not even known whether there are infinitely many.The main idea, due to the second author, is that the scarcity of noncyclic groups can be attributed to the fact that they have many automorphisms. This naturally leads to the heurißtic assumption that isomorphism classes G of abelian groups should be weighted with a weight proportional to l/# Aut G . This is a very natural and common weighting factor, and it is the purpose of this paper to show that the assumption above, plus another one to take into account the units, is sufficient to

Abstract. This paper 19 devoted to the deacnption and analysis of a new algonthm to factor positive mtegers It depends on the use of elliptic curves The new m et b öd α obtained from Pollird's p-1-method (Proc Cambridge Philos Soc 76 (187-1), 521-528) by replacing the multiplicative group by tbe group of points on a random elliptic curve 1t u conjectured thit the algonthm determmes a non-tnvial divuor of a composite number n m expected time at most K(p)(logn)2, where p is the least pnme dmding n and K u a function for which logK(t)=v'(2+o(l))logiloElogz for t->oo In the worst case, when n la the product of two pnmes of the same order of magnitude, this is exp((l+o(l))\/lognloglogn) (for n->oo) There are several other factonng algonthms of which the conjectural expected running time is given by the latter formula However, these algonthms have a running time (hat is basically lodependent of the size of the pnme factors of n, whereas the new elliptic curve method is subatantially faster for amall p