FACTORING BIVARIATE SPARSE (LACUNARY) POLYNOMIALS

FACTORING BIVARIATE SPARSE (LACUNARY) POLYNOMIALS Abstract. We resent a deterministic algorithm for comuting all irreducible factors of degree d of a given bivariate olynomial f K[x, y] over an algebraic number field K and their multilicities, whose running time is olynomial in the bit length of the sarse encoding of the inut and in d. Moreover, we show that the factors over Q of degree d which are not binomials can also be comuted in time olynomial in the sarse length of the inut and in d. Introduction Effective factorization of olynomials, when ossible, is an imortant task in comutational algebra and number theory. This roblem has a long history, going back to I. Newton in 1707, and to the astronomer F. von Schubert who in 1793 resented an algorithm for factoring a univariate olynomial, later rediscovered and generalized by L. Kronecker in 1882. Many other more efficient algorithms were designed since then: we cite [Zas69], based on [Ber70], among the most famous ones. In 1982, A.K. Lenstra, H.W. Lenstra Jr. and L. Lovász made a fundamental advance by obtaining the first deterministic olynomial-time algorithm for factoring a univariate olynomial over the rationals. Based on [LLL82] and the technique of lattice basis reduction introduced for its roof, several new factorization algorithms were obtained [CG82, Len84, Kal85, Lan85, Len87, Lec05, BHKS05]. These algorithms succeeded in bringing to olynomial time the roblem of factoring univariate and multivariate olynomials over algebraic number fields when given by their dense encoding, that is the inut f is given by the list of all its terms of degree deg(f) including the zero ones. For ractical uroses, it is worth considering the sarse (or lacunary) encoding of a olynomial. In this aer we consider the roblem of factoring a bivariate olynomial t f = a i x αi y βi Q[x, y] i=1 given in sarse encoding, i.e. by the list (a i, α i, β i ) 1 i t of its non-zero coefficients and corresonding exonents. Let l(f) denote the bit length of the sarse encoding of f ; informally seaking this is the number of bits needed to sell out the data. We obtain a deterministic algorithm for comuting the low degree factors of f in time olynomial in l(f) : Theorem 1. There is a deterministic algorithm that, given f Z[x, y] and d 1, comutes all irreducible factors of f in Q[x, y] of degree d together with their multilicities, in (d l(f)) O(1) bit oerations. 2000 Mathematics Subject Classification. Primary 11Y05; Secondary 11Y16, 11G50. Key words and hrases. Polynomial factorization, lacunary (sarse) olynomials, height of oints, Lehmer s roblem. M. Avendaño was suorted by a CONICET fellowshi, Argentina. T. Krick was artially suorted by research grants UBACYT X-112 and CONICET PIP 2461/01, Argentina. M. Sombra was suorted by the Ramón y Cajal rogram of the Ministerio de Educación y Ciencia, Sain. 1

2 Actually, this algorithm alies for factoring bivariate olynomials over number fields (see Subsection 3.2). Let us observe that the degree of a olynomial can be exonentially big in its sarse length: we have deg(f) 2 l(f) and this uer bound is attainable. A direct alication of the algorithms for factoring dense olynomials would give an exonential comlexity. The restriction to bounded degree factors is unavoidable: the olynomial f = x 1 ( rime) is of sarse length log 2 () + O(1) but has the dense irreducible factor x 1 + + 1. The first result in this direction aeared in 1998, when F. Cucker, P. Koiran and S. Smale showed how to find all the integer roots of a univariate olynomial with integer coefficients in olynomial time in its sarse encoding, and asked whether one can find in the same time the rational roots as well [CKS99]. This question (and more!) was affirmatively answered by H.W. Lenstra Jr. who resented an algorithm that given a number field K and a univariate olynomial f K[x] comutes all its irreducible factors of degree d together with their multilicities, in (d+l(f)) O(1) bit oerations [Len99b, Thm]. The first and insiring result in the multivariate setting was obtained by E. Kaltofen and P. Koiran [KK05, Thm 3] last year, who showed how to comute the linear factors of a bivariate olynomial f Q[x, y] in olynomial time in l(f). Our result is then an extension of Kaltofen-Koiran s, and a full generalization of Lenstra s to the case of bivariate olynomials. All these algorithms (including ours) are based on a ga rincile first alied by Cucker, Koiran and Smale. The idea is so strikingly simle and natural that it deserves to be exlained. Let f Z[x] and ξ Z be given, how can we test if f(ξ) = 0? Direct evaluation is not feasible, as the size of f(ξ) can be exonentially big in the inut size; an imortant excetion to this are the easy cases ξ = 0, ±1. For the other cases, assume that f = t i=1 a ix α i can be slit as f = r + x u q for non-zero olynomials r of degree deg(r) = k and q, where there is a ga between the exonents of r and those of x u q of length u k log 2 f 1 (here f 1 := t i=1 a i denotes as usual the l 1 -norm of f ). Then, excet for the cases ξ = 0, ±1, this imlies that f(ξ) = 0 if and only q(ξ) = r(ξ) = 0 : if this were not the case, namely f(ξ) = 0 but q(ξ) 0, then r(ξ) r 1 ξ k < f 1 ξ k and r(ξ) = ξ u q(ξ) ξ u so that f 1 > ξ u k 2 u k, which contradicts the ga assumtion! Therefore, to test if f vanishes at ξ 0, ±1, one decomoses f into widely saced short ieces f = i x i f i and tests if f i (ξ) = 0 for all i. One crucial fact here is that the decomosition is indeendent of the oint ξ ; therefore to find integer roots it is enough to find the common roots of a set of low degree olynomials. The other key ingredient that makes the above argument work is that any integer ξ 0, ±1 satisfies a uniform lower bound ξ 2! In order to aly the same idea to ξ Q, the correct generalization of the absolute value is the height, defined as the maximum between relatively rime exressions for the numerator and denominator. By imitating the argument above, but this time for the usual absolute value and all the -adic ones, we arrive at the same conclusion as a consequence that all rational numbers excet 0, ±1 have height at least 2. This is essentially what Lenstra alied in [Len99b]; more generally, he was able to handle in this way other factors besides the

FACTORING BIVARIATE SPARSE (LACUNARY) POLYNOMIALS 3 linear ones by considering the height of their roots after alying a suitable lower bound for them, namely Dobrowolski s theorem [Dob79] in the version of P. Voutier [Vou96]. In [KK05], the authors succeeded to resent the first generalization of this ga rincile for non-univariate olynomials, more recisely for linear factors of bivariate olynomials. As in these revious works, the key of our algorithm is a suitable ga theorem. We obtain it as a consequence of a lower bound for the height of Zariski dense oints lying on a curve due to F. Amoroso and S. David [AD00], as exlained in detail in Section 2. This result allows to decomose the given olynomial f Q[x, y] into short ieces; the factors of f are then comuted as the common factors of these low degree ieces. This strategy works for all factors excet the trivial x and y and the cyclotomic ones, that is, factors which are a roduct of binomials (including monomials) whose coefficients are roots of the unity. As in the univariate and linear bivariate cases, these factors have to be handled searately, see Section 3. Since our algorithm oerates by reducing to the cases of dense bivariate and sarse univariate olynomials, our concern is only to rove that this reduction can be done in olynomial time in the sarse encoding. We have not attemted to comute the exonent in the comlexity estimate, which in rincile can be quite big. It is certainly ossible to imrove it in view of ractical imlementation: in Subsection 3.4 we resent one idea in this direction, which consists on adating the decomosition of f to the size of the candidate factor. As a consequence of the algorithm, we derive that the number of irreducible factors of degree d of f Q[x, y] counted with multilicities (different from the trivial factors x or y ) is bounded by (d l(f)) O(1). This is not trivial, as the degree of f can be exonential in l(f), but in fact much better can be said: Proosition 2. Let f Z[x 1,..., x n ] and consider the factorization f = q where q is a cyclotomic olynomial, Q[x 1,..., x n ] runs over all non-cyclotomic irreducible factors of f, and e is the corresonding multilicity. Then e 5 6 n 3 log f 1 log 3 (8n deg(f)). e In articular the total number of non-cyclotomic irreducible factors of any degree of f is olynomially bounded in terms of the sarse length of f. This fairly unexected roerty generalizes [Dob79, Thm 2] and is a further consequence of the connection with Diohantine Geometry via the theory of heights: the Amoroso-David lower bound together with the theorem of successive algebraic minima of S.-W. Zhang [Zha95] imly a lower bound for the Mahler measure of a non-cyclotomic olynomial, and from this the statement follows easily. Moreover, a ositive answer to Lehmer s roblem would imly in the univariate case, see Subsection 1.2 for details, the stronger estimate e c log f 1. for some absolute constant c > 0. This is even more surrising, since the right-hand side deends on the coefficients of f but not on its degree. It would be interesting to determine if it is ossible to obtain such a bound without assuming Lehmer s conjecture. Proosition 2 should be comared with another result of H.J. Lenstra Jr.: the total number of irreducible factors of degree d of f Q[x] counted with multilicities (different from x ) is

4 bounded by c t 2 2 d d log(2dt) where t is the number of non zero terms of f [Len99a, Thm 1]. This bound is exonential, but indeendent of the degree and coefficients of f. Based on these two results, it seems natural to consider the following generalization of Descartes rule of signs: is the number of all irreducible (and non-cyclotomic maybe?) factors different from x of a t -nomial in Q[x] uniformly bounded by some function B(t) deending only on t, and maybe even by t O(1)? Trying to get further, one might ask if it is ossible to comute in olynomial time the absolute factorization of a olynomial given in sarse encoding, that is, its irreducible factors over Q. For the univariate case the answer is clearly no : a univariate olynomial slits comletely as a roduct of linear factors, and this cannot be done in sarse olynomial time. For the bivariate case, it can be shown that the comutation of binomial factors is equivalent to the factorization of a univariate olynomial, so that binomials factors over Q cannot be comuted either. Here, we show that excet for these, we can comute all other irreducible factors over Q of low degree, in sarse olynomial time. To give sense to such a statement, we have to secify the way algebraic coefficients are handled: a number field K is described by an irreducible monic olynomial g = δ 1 j=0 g jz j Z[z] such that K = Q(θ) for one of its roots, and this g is given in dense reresentation by the list of all coefficients g j in some secified order, including the zero ones. Each irreducible factor in the outut of the algorithm is encoded by giving a number field K such that K[x, y] and by the dense list of its coefficients, each coefficient b K being reresented by its vector of rational comonents b := (b 0,..., b δ 1 ) with resect to the basis (θ j ) 0 j δ 1. Theorem 3. There is a deterministic algorithm that, given f Q[x, y] and d 1, comutes all irreducible factors of f in Q[x, y] of degree d, together with their multilicities, excet for the binomial ones, in (d l(f)) O(1) bit oerations. This algorithm follows from another suitable ga theorem that we obtain as a consequence of a further result by Amoroso and David, a quantitative version of the Bomogolov roblem over the torus [AD03]. Furthermore, we deduce from their result an estimate for the number of non-binomial factors of a given f Q[x 1,..., x n ] (Proosition 1.4). Several interesting questions arose during our work. The most obvious is the extension of these algorithms to multivariate olynomials; this seems quite feasible as the necessary lower bounds for the height of oints in a hyersurface already aeared in the literature [AD00, AD03, Pon01, Pon05b]. An interesting oen roblem is the following: the restriction to comuting bounded degree factors kees their length under control, giving the ossibility of comuting them in sarse olynomial time. But, what if we look for factors with a fixed number of monomials, can we still find all of them in sarse olynomial time? For instance, can we comute all trinomial factors of a given f Q[x] in olynomial time? = a 1 x α 1 + a 2 x α 2 + a 3 x α 3 Q[x] The outline of the aer is as follows. In Section 1 we exlain the basics of the height theory for oints, olynomials and curves, and we rove the uer bounds for the number of factors of a sarse olynomial. In Section 2 we obtain the ga theorems, as a consequence of the lower bounds for the height of oints on curves. In Section 3 we resent the algorithms for rational and absolute factorization and estimate their theoretical comlexity.

FACTORING BIVARIATE SPARSE (LACUNARY) POLYNOMIALS 5 Note. Theorem 1 was indeendently achieved in [KK06] by Kaltofen and Koiran. This article also relies on the method in [CKS99], [Len99b] and [KK05], although it differs from ours in all other asects: the corresonding ga theorem is obtained as a consequence of a lower bound for the height of numbers in abelian extensions due to Amoroso and F. Zannier, and the binomial factors are handled differently. As observed by the authors, their algorithm requires the a riori knowledge of a universal but non-exlicit constant c [KK06, Thm 1]. In the resent aer this roblem is avoided by using the more exlicit results in [AD00] and [Pon01]. Our aroach also allows to comute not only the rational factors but also the absolute ones. Acknowledgements. We thank Corentin Pontreau for helful discussions on lower bounds for the height. The core of this aer was written during October December 2005 while M. Sombra was visiting the University of Buenos Aires, Argentina; he articularly thanks Ricardo Durán for his invitation. He also thanks the Mathematical Sciences Research Institute at Berkeley, USA, where he stayed during January 2006. 1. Heights Throughout this aer Q denotes the field of rational numbers, K a number field, L a finite extension of K, Q an algebraic closure of Q and G the subset of Q of all roots of the unity. We denote by A n the affine sace of n dimensions over Q. For a olynomial Q[x 1,..., x n ] we denote by Z() A n the affine hyersurface defined by. A curve or a variety is assumed to be equidimensional; by irreducibility of a variety we understand its geometric irreducibility, that is with resect to Q. For every rational rime we denote by the -adic absolute value over Q such that = 1. We also denote the ordinary absolute value over Q by or simly by. These form a comlete set of indeendent absolute values over Q : we identify the set M Q of these absolute values with the set {, ; rime}. More generally, we write M K for the set of absolute values over K extending the absolute values in M Q, and we note by MK the subset of Archimedean absolute values of M K. For v 0 M Q we denote by Q v0 the comletion of Q with resect to the absolute value v 0. In case v 0 = we have Q = R, while in case v 0 = is a rime, Q is the -adic field. There exists a unique extension of v 0 to an absolute value over the algebraic closure Q v0. For v M K we also denote by K v the comletion of K with resect to v. If v extends an absolute value v 0 M Q, then K v is a finite extension of Q v0. We denote σ v : K Q v a (not necessarily unique) embedding corresonding to v, that is such that a v = σ v (a) v0 for every a K. 1.1. Height of oints and olynomials. In this subsection we introduce the basic definitions and roerties of the height of oints and olynomials that we will use in the sequel. We refer for instance to [HS00] for a comlete treatment. The (logarithmic) height h(ξ) of an algebraic number ξ Q can be defined in terms of its rimitive integer minimal olynomial ξ (x) = c (x σ(ξ)) Z[x] σ:k Q where σ runs over all Q -embeddings of K := Q(ξ) in Q, by the formula (1) 1 h(ξ) = log c + [K : Q] max{0, log σ(ξ) }. σ:k Q

6 We have h(ξ) 0, and h(ξ) = 0 if and only if either ξ = 0 or ξ G, the subset of Q of all roots of 1 (Kronecker s theorem). Besides, for a rational ξ = m/n Q in reduced exression, we easily check that h(ξ) = log max{ m, n}. Alternatively, the height can be defined via the Mahler measure of the minimal olynomial as m( ξ ) := 1 this identity is a consequence of Jensen s formula. 0 log ξ (e 2πiu ) du = [K : Q] h(ξ); More generally, the height of a oint ξ := (ξ 1,..., ξ n ) A n h(ξ) := 1 [K : Q] is defined via the Weil formula v M K [K v : Q v ] log max{1, ξ 1 v,..., ξ n v } for any number field K containing the coordinates ξ i. For n = 1 this gives 1 h(ξ) = [K v : Q v ] log max{1, ξ v } [K : Q] v M K and it can be shown that this coincides with the revious definition. With this exression we readily verify that for ξ, η Q we have that h(ξ η) h(ξ) + h(η) and h(ξ n ) = n h(ξ) for n Z; in articular h(ξ 1 ) = h(ξ) and h(ω ξ) = h(ξ) for any root of unity ω G. We will be mostly interested on oints in the lane ξ = (ξ 1, ξ 2 ) A 2, in that case the formula reduces to h(ξ) = 1 [K : Q] v M K [K v : Q v ] log max{1, ξ 1 v, ξ 2 v }. Now we introduce a few notions for the height of a olynomial that will rove useful in the sequel. We will restrict to bivariate olynomials, although it is clear that all this extends to the multivariate case. For a olynomial f = t i=1 a i x α i y β i K[x, y], its absolute value with resect to v M K is The height of f is then defined as h(f) := f v := max{ a 1 v,..., a t v }. 1 [K : Q] v M K [K v : Q v ] log( f v ), which is invariant by scalar multilication because of the roduct formula v M K [K v : Q v ] log( a v ) = 0, a K. Therefore h(f) is the Weil height of the rojective oint (a 1 : : a t ). This is indeendent of the chosen field K as long as it contains all of the a i s. For a bivariate olynomial with comlex coefficients f C[x, y] we consider the Mahler measure m(f) := 1 1 0 0 log f(e 2πiu, e 2πiv ) du dv,

FACTORING BIVARIATE SPARSE (LACUNARY) POLYNOMIALS 7 and for a olynomial f K[x, y] with algebraic coefficients we define its (global) Mahler measure by the adelic formula 1 m Q (f) := [K v : Q v ] m(σ v (f)) + [K v : Q v ] log f v. [K : Q] v M K We also consider the height associated to the l 1 -norm: 1 h 1 (f) := [K v : Q v ] log σ v (f) 1 + [K : Q] v M K v / M K v / M K where for v M K, the usual definition σ v(f) 1 := i σ v(a i ) holds. For a rimitive f Z[x, y], these notions give [K v : Q v ] log f v, h(f) = log f = log max{ a 1,..., a t }, h 1 (f) = log f 1 = log( a 1 + + a t ), m Q (f) = m(f). All these are invariant by scalar multilication. In general for any f Q[x, y] write f = c f for some c Q and f Z[x, y] the rimitive olynomial with integer coefficients associated to f, then h(f) = log f, h 1 (f) = log f 1 and m Q (f) = m( f). We will use the following comarison between the heights of a given f K[x, y], which can be directly roven from the definitions: (2) h(f), m Q (f) h 1 (f) h(f) + log(t). 1.2. Height of and on lane curves. A lane curve C A 2 can have some isolated oints of small height. For instance the line Z(x + y 1) A 2 has the oints (1, 0), (0, 1), ( (1 ± 3)/2, (1 3)/2 ) all of whose coordinates are roots of 1 and so their height is 0. D. Zagier [Zag93] showed that the height of any other oint ξ Z(x + y 1) is bounded from below by a ositive constant h(ξ) h(ξ 0 ) = 0.1911 where ξ 0 denotes the largest real root of the olynomial x 6 x 4 1. Somehow the fact that a curve has some torsion oints on it does not reflect its general behavior. A more interesting arameter is the height of a Zariski dense set of oints. This is measured by the essential minimum, which for a lane curve C A 2 is defined as µ ess (C) := inf { η 0 : {ξ C : h(ξ) η} is an infinite set }. For instance, thanks to Zagier s result, µ ess (Z(x + y 1)) 0.1911. This is a articular case of the Bogomolov roblem over the torus roved by Zhang [Zha95] which asserts that for a subvariety of T n := (Q ) n, the vanishing of the essential minimum is equivalent to being torsion. This result, and others we are going to use, are stated for the torus, but T n is naturally embedded as an oen subset of A n, and since these results deend on Zariski dense sets, they can all be translated to A n. For an irreducible lane curve C A 2, being torsion is equivalent to say that there exist α, β 0 not both zero, and ω G {0} such that either C = Z(x α ωy β ) or C = Z(x α y β ω).

8 The irreducible curve C is (we should rather say corresonds to ) a translate of a subgrou whenever there exists ξ Q such that either C = Z(x α ξy β ) or C = Z(x α y β ξ). By definition, a general affine lane curve is torsion (res. translate of a subgrou) if and only if all its irreducible comonents are so. The statement of the Bogomolov roblem (now a theorem) is that µ ess (C) = 0 if and only if C is torsion. In other words, if C is not of this form, there exists a ositive constant c(c) > 0 such that h(ξ) c(c) for all but a finite number of ξ C. There is an extension of the notion of Weil height of oints to higher-dimensional varieties. This notion was first introduced by P. Philion [Phi91]; for an irreducible hyersurface V A n defined by a olynomial K[x 1,..., x n ], it coincides with the global Mahler measure of [DP99, Pon01]: (3) h(v ) = m Q (). The distribution of the height of algebraic oints in a curve is in close connection with the height of the curve itself. The relation is given by the theorem of algebraic successive minima of Zhang [Zha95, Thm 5.2 and Lem. 6.5(3)]: µ ess (C) h(c) deg(c) 2µess (C). Actually, Zhang s result is more recise (all successive minima aear, not only the first one which is the essential minimum) and more general, as it works for varieties of any dimension and for any reasonable height function. The stated version is sufficient for our alication; for a more elementary roof we refer to [DP99, 6]. It is an oen roblem to determine if this estimate is otimal for the case of lane curves or more generally for hyersurfaces ( it has been shown to be otimal if we allow varieties of higher codimension [PS04, Thm 5.1]). Thanks to this result, the Bomogolov roblem for lane curves can be rehrased as h(c) = 0 if and only if C is torsion. Under this form, the conjecture was already roven by W. Lawton in 1977 [Law77]. For ξ Q we have that h(ξ) = 0 if and only if ξ G ; this is the 0-dimensional (easy) case of the Bogomolov roblem. Lehmer s conjecture gives a lower bound for the height of non-torsion oints, its statement being that there exists a ositive constant c > 0 such that c h(ξ) for ξ / G. [Q(ξ) : Q] This conjecture has been widely generalized. Here we are only interested in the case of curves: Conjecture 1.1. (i) Lehmer s roblem for lane curves: Let C A 2 be an irreducible curve defined over a number field K which is not torsion. Then there exists a universal c > 0 such that µ ess c (C) [K : Q] deg(c). (ii) Effective Bogomolov roblem for lane curves: Let C A 2 be an irreducible curve which is not a translate of a subgrou. Then there exists a universal c > 0 such that µ ess c (C) deg(c).

FACTORING BIVARIATE SPARSE (LACUNARY) POLYNOMIALS 9 These two conjecture look similar but they are not. The generalization of Lehmer s roblem is of arithmetic nature since the degree of the number field lays a role, while the quantitative Bogomolov roblem is of geometric nature since it makes no reference to the field of definition. It has been shown that conjecture 1.1(i) is imlied by the classical Lehmer s roblem [Law77]. Conjecture 1.1(ii) is [DP99, Conj. 1.1]. Because of the theorem of successive minima, it is equivalent to have lower bounds for the essential minimum or for the height, that is the (global) Mahler measure of the defining olynomial of C. Nowadays all these results are roved u to an ε : for the Lehmer s roblem we will be mainly alying the following lower bound due to Amoroso and David [AD00], in the version of C. Pontreau [Pon05a, Pro. IV.1] who simlified the roof and made all constants exlicit: if C A 2 is a non-torsion curve defined by an irreducible olynomial Z[x, y] of degree d, then (4) µ ess (C) 1 ( ) 3 log log(16d) 5 6 d. log(16d) In the reference this result is stated in terms of h(c) ; you have to look into the roof for the version u here. In fact we will be using the version over a number field: Corollary 1.2. Let C A 2 be a curve defined by an irreducible olynomial K[x, y] which is not of the form = i (xα ω i y β ) nor = i (xα y β ω i ) for some α, β 0 not both zero and ω i G {0} and set d := deg(c) = deg(). Then ( ) 3 µ ess 1 log log(16[k : Q]d) (C) 5 6 [K : Q]d. log(16[k : Q]d) This follows immediately from (4) by considering the norm N() := σ() Q[x, y]. σ:k Q For the effective Bogomolov roblem we use another result of Amoroso and David: if C A 2 is a curve which is not a translate of a subgrou and d := deg(c) = deg(), then [AD03, Thm 1.5]: (5) µ ess (C) 1 (log log(d + 2))4 2 70 d (log(d + 2)) 5. 1.3. On the number of factors of a sarse olynomial. General lower bounds for the Mahler measure immediately yield uer bounds for the number of factors of a given olynomial. To the best of our knowledge, this observation aears for the first time in the work of E. Dobrowolski [Dob79]. Here we treat the general n -dimensional case. The notions and results of the revious subsection extend to hyersurfaces. We will state them but instead refer the interested reader to the literature for n 3. We recall that a olynomial is cyclotomic if it is a roduct of binomials (including monomials) whose coefficients are roots of the unity. Proosition 1.3. Let f K[x 1,..., x n ] and consider the factorization f = q e where q is cyclotomic, K[x 1,..., x n ] runs over all non-cyclotomic irreducible factors of f, and e is the corresonding multilicity. Then e 5 6 n 3 [K : Q] h 1 (f) log 3 (8n[K : Q] deg(f)).

10 Proof. We have that m Q (q) = 0 as q is cyclotomic an so e m Q () = m Q (f) h 1 (f). For each non-cyclotomic factor Q[x 1,..., x n ] we minorate the Mahler measure by the Amoroso- David s lower bound in the version of Pontreau [Pon01, Thm 1.6] (see the estimate (4) above for the case n = 2 ), from which we derive that if V A n is an hyersurface defined by an irreducible olynomial over K, then [K : Q] h(v ) 1 5 6 n 3 ( ) 3 log(n log(8n[k : Q] deg(v )). log(8n[k : Q] deg(v )) Therefore, by Identity (3), we have 1 [K : Q] m Q () 5 6 n 3 log 3 (8n[K : Q] deg()) 1 5 6 n 3 log 3 (8n[K : Q] deg(f)), which imlies from where we deduce our result. 1 [K : Q] h 1 (f) 5 6 n 3 log 3 (8n[K : Q] deg(f)) This is a generalization to n 2 of [Dob79, Thm 2]. As said, a ositive answer to the classical Lehmer s roblem would imly a ositive lower bound for the Mahler measure of an arbitrary non-cyclotomic olynomial K[x 1,..., x n ], of the form c m Q () [K : Q] for some universal constant c > 0, namely Conjecture 1.1(i). Alying this to the argument above, the revious roosition would imrove to (6) e c 1 [K : Q] h 1 (f). In a similar way, we can roduce an uer bound for the number of non-binomial irreducible factors over Q : Proosition 1.4. Let f Q[x 1,..., x n ] and consider the factorization f = q were q is a roduct of binomials, Q[x 1,..., x n ] runs over all non-binomial irreducible factors of f, and e is the corresonding multilicity. Then e 10 14 n 8 h 1 (f) log 5 (max{16, n deg(f)}). Proof. We have that e e m Q () = m Q (f) h 1 (f); aly the Amoroso-David quantitative Bogomolov roblem in the version of Pontreau [Pon05b, Thm 1.5] (or (5) above for the case n = 2 ). Similarly, a ositive answer to the effective Bogomolov roblem (Conjecture 1.1(ii)) would imly that e c 1 h 1 (f) for a universal constant c > 0. e

FACTORING BIVARIATE SPARSE (LACUNARY) POLYNOMIALS 11 2. Ga theorems By a ga theorem, following [CKS99, Len99b, KK05], we understand a statement asserting that for a olynomial f decomosed as f = r + s for non-zero olynomials r and s, then f has a given roerty if and only r and s have it, rovided that r and s are sufficiently searated. We introduce some notation: Definition 2.1. For Q[x, y] such that deg y () 1 we set λ() := inf { η 0 : {(ω, ν) G Q : (ω, ν) = 0, h(ν) η} is an infinite set }. Since deg y () 1, for all but a finite number of ω G there exists some ν Q such that (ω, ν) = 0 and so λ() is well-defined and non-negative. In what follows we deal with irreducible olynomials, that are defined u to a scalar factor. For simlicity we always refer to one (obvious) reresentant in each class of associate irreducible olynomials. The following is the main result of this section: Theorem 2.2. Let f, r, q Q[x, y] be such that f = r + y u q. Let also be given an irreducible olynomial Q[x, y], y, such that deg y () 1, and suose that (u deg y (r)) λ() h 1 (f). Then divides f if and only if it divides r and q. For its roof we need the following lemma: Lemma 2.3. Let f, r, q Q[x, y] be such that f = r + y u q. Let also be given ω G and ν Q be such that f(ω, ν) = 0 but q(ω, ν) 0. Then there exists a constant δ(f) > 0 not deending on (ω, ν) such that (u deg y (r)) h(ν) h 1 (f) δ(f). Proof. Let K be a number field containing the coefficients of f, ω and ν, and set k := deg y (r). For each absolute value v M K we have two cases: ν v 1 : since ω v = 1 we have that { σv (q) 1 for v MK q(ω, ν) v, q v for v / MK. ν v > 1 : using that f(ω, ν) = r(ω, ν) + ν u q(ω, ν) = 0 we infer that { ν k ν u v σ v (r) 1 for v MK v q(ω, ν) v = r(ω, ν) v, ν k v r v for v / MK. As both r and q are non-zero, σ v (q) 1, σ v (r) 1 < σ v (f) 1 and so log σ v (q) 1, log σ v (r) 1 log σ v (f) 1 δ(f) for some δ(f) > 0 deending only on f. The revious inequalities imly that { log σv (f) 1 δ(f) for v MK (u k) log max{1, ν v } + log q(ω, ν) v, log f v for v / MK.

12 By summing u over all absolute values, using the roduct formula and the definition of the height, one obtains that 1 ( ) (u k) h(ν) = [K v : Q v ] (u k) log max{1, ν v } + log q(ω, ν) v [K : Q] v M K 1 [K v : Q v ] ( log σ v (f) 1 δ(f) ) + [K v : Q v ] log f v [K : Q] v M K v / M K = h 1 (f) δ(f). Proof of Theorem 2.2. The is trivial, so we show the other imlication. Suose that f but q. From the fact that is irreducible we have that the set of common roots of and q is finite. Also, since deg y () 1 and y, the set {(ω, ν) G Q : (ω, ν) = 0} is infinite. Given ε > 0, it follows from the definition of λ() that the set {(ω, ν) G Q : (ω, ν) = 0, h(ν) λ() ε} is finite. Therefore there exist an infinite number of (ω, ν) G Q such that (ω, ν) = 0 and h(ν) > λ() ε, and there still exist some ω G and ν Q such that Alying Lemma 2.3 Since this holds for all ε > 0, we infer (ω, ν) = 0, q(ω, ν) 0 and h(ν) > λ() ε. (u k) (λ() ε) (u k) h(ν) h 1 (f) δ(f). (u k) λ() h 1 (f) δ(f) < h 1 (f) because δ(f) does not deend on (ω, ν) and so does not deend on ε either. This contradicts the hyothesis: (u k) λ() h 1 (f). Therefore q and y u q = r as wanted. Corollary 2.4. Let f, r, q Q[x, y] be such that f = r + y u q. Let also be given n 1 and an irreducible olynomial Q[x, y], y, such that deg x () 1, deg y () 1, and suose that (u deg y (r)) λ() h 1 (f) + (n 1) log(deg x (f)). Then n divides f if and only if it divides r and q. Proof. Since deg x () 1, we have that n f if and only if j f x j for j = 0,..., n 1. The result follows by alying the Ga Theorem 2.2 to j f/ x j : We have j f x j = j r x j + yu j q x j. If j r/ x j or j q/ x j vanish, there is nothing to rove. Otherwise, u deg y ( j r/ x j ) u deg y (r) since deg y ( j r/ x j ) deg y r. Furthermore, from the definition of h 1 we infer that h 1 ( j f x j ) h 1(f) + (n 1) log(deg x (f)), since for a coefficient a i of f in K and d N, for v M K, σ v( f/ x) 1 deg x f σ v (f) 1 holds, while for v / M K, f/ x v f v since k v = 1 for k N.

FACTORING BIVARIATE SPARSE (LACUNARY) POLYNOMIALS 13 We observe that for instance by [Len99b, Pro. 3.2], we know the a riori bound n t 1. Of course this result is only useful whenever λ() > 0. What haens is that this arameter is bounded from below by the essential minimum, and so all existing estimations for the essential minimum will give us a corresonding ga theorem. Lemma 2.5. Let be an irreducible olynomial in K[x, y] such that deg y () 1. Then λ() µ ess (Z()). Proof. Observe that h(ν) = h(ω, ν) ; we can then rehrase the definition of λ() as λ() = inf { η 0 : {ξ Z() (G Q) : h(ξ) η} is an infinite set }. Comare with the definition of the essential minimum: µ ess (Z()) = inf { η 0 : {ξ Z() : h(ξ) η} is an infinite set }, so that λ() is the infimum over a subset of the set used to define µ ess (Z()) and the inequality is clear. Equality in Lemma 2.5 above does not necessarily hold: consider := x α ξy β, then for any (ω, ν) G Q we have that (ω, ν) = 0 ν β = ω α /ξ and so Hence h(ν) = h(νβ ) β = h(ωα /ξ) β = h(ξ) β. λ((x, y)) = h(ξ)/β while λ((y, x)) = h(ξ)/α. In articular, λ deends on the order of the variables, while of course the essential minimum does not, so there cannot coincide in general. One can rove, however, that µ ess () = h(ξ)/ max{α, β} [PS04, Pro. 5.4]. From Corollary 1.2 we deduce: Corollary 2.6. Let f, r, q K[x, y] be such that f = r + y u q. Let also be given n t 1 and an irreducible olynomial K[x, y], deg x 1, that is non-cyclotomic, that is, not of the form = i (xα ω i y β ) nor = i (xα y β ω i ) for some α, β 0 not both zero and ω i G {0}, and set d := deg(). Suose that u deg y (r) 5 6 [K : Q] d log 3 (16[K : Q]d) (h 1 (f) + (t 2) log(deg x (f))). Then n divides f if and only if it divides r and q. Similarly we obtain the following ga theorem from the lower bound (5): Corollary 2.7. Let f, r, q Q[x, y] be such that f = r + y u q. Let also be given n t 1 and an irreducible olynomial Q[x, y] which is not a binomial, and set d := deg(). Suose that u deg y (r) 2 70 d log 5 (d + 2) (h 1 (f) + (t 2) log(deg x (f))). Then n divides f if and only if it divides r and q.

14 3. Comuting the low degree factors of sarse olynomials The goal of this section is to resent the rational and absolute factorization algorithms for sarse bivariate olynomials. Our conventions about encoding are the usual ones, the same as in for instance [Len99b]. The number of bits needed to write down a non-zero integer a Z is log 2 (a) + 1 for the digits and 1 more for the sign. For a rational a = m/n Q in reduced exression, we define its bit length as l(a) = l(m) + l(n) 2 = log 2 m + log 2 (n) + 2; the somewhat artificial 2 is there just to make this coincide with the revious notation for an integer a. The sarse encoding of f = t i=1 a ix αi y βi Q[x, y] is the list (a i, α i, β i ) 1 i t of its (non-zero) coefficients and corresonding exonents, and so its bit length is t ( ) (7) l(f) := l(a i ) + log 2 (α i ) + log 2 (β i ) + 2 ; i=1 observe that l(f) is an uer bound for t, log 2 (deg f) and h(f), and in fact is olynomially equivalent to these quantities: l(f) = (t log 2 (deg f) h(f)) O(1). For encoding olynomials over number fields we have to say how number fields and algebraic numbers are handled: a number field K of degree δ = [K : Q] is described by an irreducible monic olynomial g = δ 1 j=0 g jz j Z[z] such that K = Q(θ) for one of its roots, and this g is given in dense reresentation by the (ordered) list of all its coefficients g j including the zero ones. The length of this descrition is δ 1 l(k) := l(g j ); j=0 in articular l(k) [K : Q], h(g). An element b K is reresented by its vector of rational comonents (b 0,..., b δ 1 ) with resect to the basis (θ j ) 0 j δ 1. It can be shown by (you need some estimate between the height of an algebraic integer and that of its minimal olynomial) that h(b) l K (b) + [K : Q](h(g) + [K : Q] log(2)) = (l(k) + l K (b)) O(1). A sarsely given olynomial f = t i=1 a ix αi y βi K[x, y] is then encoded by the list of its (non-zero) coefficients and corresonding exonents, and its length relative to K is t l K (f) := (l K (a i ) + l(α i ) + l(β i )). i=1 Note that the inut data is secified by f and K, and so the inut length is l(k) + l K (f). We have that t, log 2 (deg f) l(f) and h(f) l K (f) + [K : Q](h(g) + [K : Q] log(2)) = (l(k) + l K (f)) O(1). When the inut of our algorithms comrises an inclusion K L of number fields, L is described as an extension of K by a monic irreducible olynomial k(z) O K [z] such that L = K(ϑ) for a root ϑ of k ; this olynomial is reresented in a dense way. A olynomial L[x, y] in the outut is then encoded by the (dense) list of its coefficients with resect to the roduct basis (θ j ϑ k ) 0 j δ 1,0 k γ 1 of L over Q ; here we set γ := [L : K]. Note that for an element b K in the base field encoded as b = b 0 + + b δ 1 x δ 1 with resect to the given basis of K over Q, its encoding with resect to the roduct base will be the same and so l L (b) [L : K] l K (b) since we have to count the zero coefficients corresonding to the monomials θ j ϑ k with k 1. In articular l L (f) [L : K] l K (f) for f K[x, y].

FACTORING BIVARIATE SPARSE (LACUNARY) POLYNOMIALS 15 For the absolute factorization algorithm for f K[x, y], the outut irreducible olynomials i Q[x, y] are encoded by (L i, i ), where L i consists in the minimal extension of K such that i L i [x, y] (we observe that this encodes a full set (σ( i )) σ:k Q of [L i : K] conjugate factors of f ). The coule (L i, i ) is encoded by a monic irreducible olynomial k i (z) O K [z] such that L i = K[z]/(k i (z)), and i is given by its coefficients. 3.1. Binomial factors. The comutation of the irreducible factors of a bivariate olynomial that are binomials or, more generally, roducts of binomials can be reduced to the univariate case as we show in this section. We first observe that if an irreducible olynomial K[x, y] is a roduct of binomials then it has one of the following forms: (8) (x, y) = σ (x α σ(ξ)y β ) or (x, y) = σ (x α y β σ(ξ)), where α, β 0 are not 0 simultaneously, ξ Q and where σ : K(ξ) Q runs over all K -embeddings of K(ξ) in Q. We have the following results: Lemma 3.1. Let α, β, n N, ξ Q and f Q[x, y] be given. Set z for a new variable and denote by g Q[x, y, z] the remainder of the division with resect to the variable x of f(x, y) by the monic olynomial x α zy β. Then Proof. Consider the ring (x α ξ y β ) n f(x, y) (z ξ) n g(x, y, z). A := Q[x, y ±1, z]/(x α zy β ). We have that x α ξy β = (z ξ)y β in A, and, since y is invertible, we have the following equality of ideals ((x α ξy β ) n ) = ((z ξ) n ) in A. We call this ideal I. By definition f = g in A and so f I if and only if g I, that is (x α ξ y β ) n f(x, y) in A (z ξ) n g(x, y, z) in A. We have to show that we can take out the words in A from the above statement. We observe that there is a natural identification A = Q[x, y ±1 ]. Therefore, (x α ξ y β ) n f in A (x α ξ y β ) n f in Q[x, y ±1 ] (x α ξ y β ) n f in Q[x, y] since y is rime to x α ξ y β. We have a second identification and therefore α 1 A = Q[y ±1, z] x j, j=0 (z ξ) n g in A (z ξ) n g in Q[x, y ±1, z] (z ξ) n g in Q[x, y, z] since y is rime to z ξ. Corollary 3.2. With the same notations than in the revious lemma, let K be a number field and suose that f K[x, y]. Set (x, y) := σ (x α σ(ξ)y β ) K[x, y] and q(z) := σ where σ runs over all K -embeddings of K(ξ) in Q. Then (x, y) n f(x, y) q(z) n g(x, y, z). (z σ(ξ)) K[z]

16 Proof. The olynomials x α σ(ξ)y β for different σ s are relatively rime, and the same is true for the olynomials z σ(ξ). Hence (x, y) n f(x, y) if and only if (x α σ(ξ)y β ) n f(x, y) for all σ if and only if (z σ(ξ)) n g(x, y, z) for all σ if and only if q(z) n g(x, y, z). The algorithm to comute the irreducible factors of f K[x, y], of degree bounded by d, that are roduct of binomials is now clear: We are looking for factors (x, y) K[x, y] of degree d of one of the forms in (8). The cases ξ = 0, α = 0 or β = 0 reduce directly to the univariate case where we aly Lenstra s algorithm [Len99b, Thm] to the corresonding content of f. So we can restrict ourselves to the cases when ξ Q and α, β N. We consider first the factors of the first form in (8). We fix 1 α, β d, and we set g := g α,β K[x, y, z] for the remainder of dividing f (with resect to x ) by x α zy β ( g deends only on f and α, β ). It is easy to comute g by Euclidean division: t g(x, y, z) = a i x αi mod α (z y β ) αi/α y βi, so that g is as sarse as f. We write i=1 g(x, y, z) = i,j g i,j (z)x i y j and observe that an irreducible factor q K[z] satisfies q n g q n g i,j for all i, j, where there are at most t non-zero olynomials g i,j, and each of them is as sarse as f, with coefficients obtained as the sum of at most t coefficients of f. We comute all irreducible factors q K[z] of g of degree bounded by d/ max{α, β} and their corresonding multilicities, by examining the common irreducible factors (and their multilicities) of all the g i,j s. This is done again alying Lenstra s univariate algorithm. Since the irreducible olynomial q is of the form q = σ (z σ(ξ)), the corresonding candidate factor of f is then derived as (x, y) = (y β ) deg(q) q(x α y β ), where deg() = max{α, β} deg(q) d. Before including within the list of factor, we check if it is irreducible by alying a factorization algorithm like [Len87, Thm. 3.26] or the recent imrovement in [Lec05]. Corollary 3.2 certifies that for given α, β, we obtain in this way all irreducible factors of f of degree d of the first form in (8), as well as their multilicities. For the factors in (8) of the second form, we roceed similarly, by considering the remainder g K[x, y ±1, z] of dividing f (with resect to x ) by x α y β z. We observe that the corresonding extensions of Lemma 3.1 and Corollary 3.2 hold. In this case, is derived from the factor q K[z] of g as (x, y) = q(x α y β ). The algorithm described above yields the following result: Theorem 3.3. There is a deterministic algorithm that, given f K[x, y] and d 1, comutes all irreducible factors of f in K[x, y] of degree d which are roducts of binomials, together with their multilicities, in ( d (l(k) + l K (f)) ) O(1) bit oerations. Proof. We have already established that the revious algorithm gives these factors and their multilicities. Its running time is estimated as follows: for each air α, β, we are alying Lenstra s algorithm t times to the olynomials g i,j of sarse length l(g i,j ) = O(l(f)), in order to comute their irreducible factors of degree d/ max{α, β} and their multilicities. This task is done in ( d (l(k) + l K (f)) ) O(1) bit oerations. Since there are at most d 2 airs α, β, the total bit cost of the algorithm remains of order ( d (l(k) + l K (f)) ) O(1).

FACTORING BIVARIATE SPARSE (LACUNARY) POLYNOMIALS 17 3.2. Rational factorization. The search of all the low degree bivariate factors of a sarse f K[x, y] is done by decomosing it as a sum of short ieces, as in the revious aers [CKS99, Len99b, KK05]. For given x, y 0, these ieces have to be searated by a distance ( ga ) of at least x in the x -direction or y in the y -direction. This is done here by decomosing f first with resect to the y -exonents, then with resect to the x -exonents. Let f = t i=1 a ix αi y βi and suose that the monomials are already ordered so that β 1 β 2 β t. Then we determine subject to the conditions l 0 := 0 < l 1 < < l s < l s+1 = t β i+1 β i < y for l j + 1 i l j+1, 0 j s, and β lj +1 β lj y for 1 j s, namely we slit the y -exonents β 1,..., β t into subsets so that consecutive exonents in the same subset are at distance < y and between different subsets there is a ga of length y. Set r j := l j+1 i=l j +1 a i x α i y β i β lj +1 for 0 j s so that f = y β l 0 +1 r 0 + y β l 1 +1 r 1 + + y β ls+1 r s. Next we do the same rocedure over each r j with resect to x : first we reorder the monomials alying a ermutation τ so that r j = l j+1 i=l j+1 a τ(i) x α τ(i) y β τ(i) β lj +1 and α τ(lj +1) α τ(lj +2) α τ(lj+1 ). Then for each 0 j s we sub-slit this set of l j+1 l j exonents into subsets such that the consecutive x -exonents in the same subset are at distance < x, and between different subsets there is a ga of length x. Using this, we decomose r j into ieces r j = x ζ0,j r 0,j + + x ζ t j,j r tj,j for some exonents {ζ i,j : 0 j s, 0 i t j } {α 1,..., α t } that we do not exlicit to avoid useless roliferation of indexes. Each r i,j is (u to a monomial) some art of r j, which in time is (u to a monomial) some art of f. We arrive in this way to a list of k t non-zero olynomials f 1,..., f k (after rewriting the r i,j s into f i s) such that (9) f = x γ 1 y δ 1 f 1 + x γ 2 y δ 2 f 2 + + x γ k y δ k f k ; and by construction for 1 i k, and for i j we have that l K (f i ) l K (f), deg x (f i ) < (t 1) x, deg y (f i ) < (t 1) y either γ j γ i deg x (f i ) x or γ i γ j deg x (f j ) x or δ j δ i deg y (f i ) y or δ i δ j deg x (f j ) y. We have decomosed f in t ieces of controlled degree and searated by a ga of length x in the x -direction or y in the y -direction. The comutation of the irreducible factors of f of degree d is then clear. Pure factors in x or y reduce to the univariate case [Len99b]. For the truly bivariate factors, we comute first a constant c such that h 1 (f) + (t 2) log(deg x (f)) c in time (l(k) + l K (f)) O(1), as in [Len99b, Pro.3.6]. We set x := y := = 5 6 [K : Q] d log 3 (16[K : Q]d) c.

18 Alying Corollary 2.6 we infer that for f = x γ1 y δ1 f 1 + x γ2 y δ2 f 2 + + x γ k y δ k f k as in (9), then for K[x, y], deg x 1, that is not a cyclotomic olynomial, we have n f n f i for all i. The rocedure consists on comuting first the cyclotomic factors together with their multilicity, by using the algorithm in Subsection 3.1. For the other factors, we comute them as the common factors of the f i s, by using any olynomial-time algorithm for factoring dense bivariate olynomials over a number field, for instance [Len87, Thm 3.26] or [Lec05]. Therefore we obtain the following result: Theorem 3.4. There is a deterministic algorithm that, given f K[x, y] and d 1, comutes all irreducible factors of f in K[x, y] of degree d, together with their multilicities, in ( d (l(k) + lk (f)) ) O(1) bit oerations. Proof. We have already established that the revious algorithm gives all these factors and their multilicities. We estimate its running time. We show that the degree of f i for all i, 1 i k, in the decomosition (9) is olynomial in the inut size. This is a consequence of our estimate for the ga length: l(f i ) l(f) and deg x (f i ), deg y (f i ) < (t 1) = O(t ([K : Q] d) 1+ε c) = ( d (l(k)+l K (f)) ) O(1). Then we aly to each f i a olynomial-time algorithm for factoring dense bivariate olynomials over K, which would do the task in ( d (l(k) + l K (f)) ) O(1) bit oerations. Since the number of f i s is at most t l, the total comlexity remains of the same order. If for an inut olynomial f K[x, y] we are interested in its factors in an extension L, we can comute them by just including f into L[x, y] and then alying the above algorithm over L ; its cost would be of ( d (l(k) + l K (f) + l K (L) ) O(1) bit oerations. We note that here, for the factors which are roducts of binomials but not cyclotomic, we have the choice of comuting them either by reduction to the univariate sarse case of Theorem 3.3 or by reduction to the dense bivariate case. 3.3. Absolute factorization. Given a olynomial f K[x, y], we can aly Corollary 2.7 to extend the revious algorithm to the comutation of all irreducible factors of f over Q, of degree bounded by d, excet the binomial ones. We assume that the inut f is encoded in K[x, y] and as before we comute a constant c such that h 1 (f) + (t 2) log(deg x (f)) c in time (l(k) + l K (f)) O(1), then we set x := y := = 2 70 d log 5 (d + 2) c. Corollary 2.7 imlies that for the associated decomosition f = x γ 1 y δ 1 f 1 +x γ 2 y δ 2 f 2 + +x γ k y δ k f k as in (9), any irreducible Q[x, y] that is not of the form satisfies (x, y) = x α ξy β or (x, y) = x α y β ξ, n f n f i for all i. Now we need to determine the common factors of the f i s over Q[x, y] and their multilicity. In order to do this, we first factor comletely each of the f i over K[x, y] by alying any dense olynomial-time bivariate factorization algorithm over K. An irreducible factor Q[x, y] of f will necessarily divide a common irreducible factor q K[x, y] of all the f i s. Thus it is enough to kee all common irreducible factors q K[x, y] of all the f i s and their multilicities, and then to factor them in Q[x, y] by alying any olynomial-time algorithm for factoring dense bivariate olynomials over Q, for instance [Kal95, Theorem 11]. We only kee those factors in the outut

FACTORING BIVARIATE SPARSE (LACUNARY) POLYNOMIALS 19 which are of degree d and which are not binomials. We roceed in this way in order to avoid comaring irreducible factors in Q[x, y] of different f i s, that can, although equal, be described in different field extensions. Theorem 3.5. There is a deterministic algorithm that, given f K[x, y] and d 1, comutes all irreducible factors of f in Q[x, y] of degree d, together with their multilicities, excet for the binomial ones, in ( d (l(k) + l K (f)) ) O(1) bit oerations. Proof. As with the revious one, the comlexity of this algorithm is estimated in ( d (l(k) + l K (f)) ) O(1) bit oerations, because we have to factor t olynomials fi of degree olynomially bounded in the inut length to find all ossible q, which are of inut length l K (q) = ( d (l(k) + l K (f)) ) O(1) and at most the same quantity, and then to factor them in Q[x, y]. 3.4. A ractical imrovement: adative ga methods. The ractical efficiency of the roosed algorithms deends essentially on the length defining the ga in f : the degree of the ieces f i deends on, and if this degree is large, the dense factorization algorithm will be clearly slower. In other words, the smaller the ga length is, the faster the algorithm works. Since the ga is roortional to the inverse of the essential minimum, the greatest the essential minimum, the faster the algorithm. There are some secial situations where we can get better bounds, for instance for linear factors (x, y) = ax + by + c with integer coefficients, as in [KK05]. The Mahler measure of a olynomial is bounded from below by the Mahler measure of any of its facet olynomials. Hence for a, b, c Z relatively rime numbers such that a b c 0, we have that m(ax + by + c) max{m(ax + by), m(by + c), m(ax + c)} = log max{ a, b, c } as it can be roved that the Mahler measure of a binomial coincides with its height. The theorem of successive minima then imlies µ ess (Z(ax + by + c)) 1 2 log max{ a, b, c } = 1 2 h(). The only case for which this lower bound is meaningless is when a, b, c = 0, ±1. (When a, b or c vanish, we reduce easily to the univariate case so we do not consider it here.) When a, b, c = ±1, Zagier s theorem [Zag93], see also Subsection 1.2, shows that h(ξ) 0.1911. Hence { log(ξ0 ) = 0.1911 if a, b, c = ±1 µ ess (Z(ax + by + c)) h() log(2) 2 = 0.3465 otherwise. which imroves the bound log(1.045) 0.0440 roosed in [KK05]. Note that in this case the ga size associated with = ax + by + c gets smaller as the coefficients of tend to infinity. Therefore, a good strategy to make the algorithm more efficient might be to exclude a finite number of candidates by testing them as factors of f (using a rough estimate for their ga length), and then use a much smaller ga length to find the rest of the factors by reduction to the dense case. References [AD00] F. Amoroso, S. David, Minoration de la hauteur normalisée des hyersurfaces. Acta Arith. 92 (2000) 339-366. [AD03] F. Amoroso, S. David, Minoration de la hauteur normalisée dans un tore. J. Inst. Math. Jussieu 2 (2003) 335-381. [BHKS05] K. Belabas, M. van Hoeij, J. Klüners, A. Steel, Factoring olynomials over global fields. Prerint (2005).

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FACTORING BIVARIATE SPARSE (LACUNARY) POLYNOMIALS 21 Deartamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires E-mail address: mavendar@dm.uba.ar Deartamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires E-mail address: krick@dm.uba.ar Deartament d Àlgebra i Geometria, Universitat de Barcelona E-mail address: sombra@ub.edu