### Finding small degree factors of lacunary polynomials [chapter]

H. W. Lenstra
Number Theory in Progress
If K is an algcbraic number field of degree at most m over thc field Q of rational numbers, and / 6 K[X] is a polynomial with dt most k non-zero terms and with /(O) / 0, then for any positive integer d the number of irreducible factors of / m K[X] of degiee at most d, counted with multiplicities, is bounded by a constant that depends only on m, k, and d This is proved m a compamon papei (H W Lenstra Jr "On the factonzation of lacunary polynomials") In the piesent paper an algonthm for actually
more » ... onthm for actually finding those factors is presented The algonthm assumes that K is specified by means oi an irreducible polynomial h with integral coefhcients and leadmg coefficient l, such that K = Q(a) for a zero α of Λ Also, the polynomial / = ]T( a%Xl ls supposed'to be given m its sparse representation, i c , äs the hst of pairs (z, o() for which a, / 0, each a, bemg represented by mcans of its vector of coefficients on the vectoi space basis l α , α^108'1^"1 of K over Q If l denotes the "length" of these mput data, when written out m bmary, then the runnmg time oi the algonthm, measured in bit operations is at most (/ + d)c for sorne absolute and effectively compulable constant c Taking K = O and d = l, one deduces that all rational zeroes of a sparsely represented polynomial with latiorial coefficients can be found m polynomial time This answers a question raised by F Gucker, P Konan, and S Smale 1991 Mathematics Subject Classification Pnmaiy 11R09, 11Y16 Key words lacunaiy polynomial, computational complexity Acknowledgernents. The author was s jpported by NSF under grant No DMS 92-24205 He thanks J A Csmk, C J Smyth, and J D Vaalei foi helpful assibtancc l. Introduction F Gucker, P Koiran, and S Smalc  exlubitod a polynomial time algonthm ac-compli&hmg the followmg Suppo&e that a polynomial f = Σι a<Xl ln one variable with cocfficicnts in thc ring Z of mtcgers is specified m its sparse representation, i c , by the h&t of pairs (i, a,) for which at 7= 0 Then the algonthm finds all zeroes of f m Z One of thc qucstions they raised is whcthei one can also find all rational /croes of / in polynomial time In thc prescnt paper I show that this is indeed the