ALGEBRAIC OSCULATION AND FACTORIZATION OF SPARSE POLYNOMIALS

POLYNOMIALS

MARTIN WEIMANN

Abstract. We prove a theorem on algebraic osculation and we apply our result to the Computer Algebra problem of polynomial factorization. We con- siderX a smooth completion ofC² andD an effective divisor with support

∂X=X\C². Our main result gives explicit conditions equivalent to that a given Cartier divisor on the subscheme (|D|,O_D) extends toX. These oscula- tion criterions are expressed with residues. We derive from this result a toric Hensel lifting which permits to compute the absolute factorization of a bivari- ate polynomial by taking in account the geometry of its Newton polytope. In particular, we reduce the number of possible recombinations when compared to the Galligo-Rupprecht algorithm.

1. Introduction

This article is originally motivated by the wellknown Computer Algebra problem of polynomial factorization. We introduce here a new method to compute the absolute factorization of sparse bivariate polynomials, which is based on criterions for algebraic osculation in toric varieties.

Since the eightees, several deterministic or probabilistic algorithms have been obtained to compute the irreducible absolute factorization of dense bivariate poly- nomials defined over a number field K ⊂ C. We refer the reader to [17] and [9]

for a large overview of the subject. In many cases, these algorithms are based on a Lifting and Recombination scheme - refered here under the generical term LR-algorithms - which mainly consists to detect irreducible absolute factors of a polynomial f ∈ K[t₁, t₂] in its formal decomposition in ¯K[[t₁]][t₂]. Although this approach a priori necessites an exponential number of possible recombinations, people succeded in the last decade to develop LR-algorithms running now in a quasi-optimal polynomial complexity (see for instance [8], [1], [9] and the reference within).

In general, a LR-algorithm first performs a generic affine change of coordinates.

Whenf has many zero coefficients in its dense degreedmonomial expansion - we say thatf issparse - this step loses crucial information. In this article, we propose a new method which avoids this generical choice of coordinates. Roughly speaking, we obtain a toric version of the Hensel lifting process (in the spirit of [1]) which detects and computes the irreducible absolute factors of f by taking in account the geometry of its Newton polytope Nf. This permits in particular to reduce the number of possible recombinations when compared to the Galligo-Rupprecht algorithm [13].

Let us present our main results.

1

Algebraic Osculation. A natural way to profit from the Newton polytope infor- mation is to embed the complex curve off in a smooth toric compactificationX of the complex plane. For a well chosenX, we can recoverNf (up to translation) from the Picard class of the Zariski closureC⊂Xof the affine curve off. The boundary

∂X=X\C²ofX is a normal crossing divisor whose Picard group satisfies P ic(∂X)'P ic(X).

Thus, it’s natural to pay attention to the restriction ofCto some effective Cartier divisorD(more precisely, to the subscheme (|D|,OD)) with support|∂X|. In order to detect the irreducible components of C, we need to find conditions for that a Cartier divisor onD extends toX. This is achieved in our main

Theorem 1. LetX be a smooth projective compactification ofC², whose boundary

∂X =X\C² is a normal crossing divisor. Let D be an effective Cartier divisor with support |∂X| and let Ω²_X(D)be the sheaf of meromorphic 2-forms with polar locus bounded byD.

There exists a pairingh·,·i between the group of Cartier divisors onD and the vector spaceH⁰(X,Ω²_X(D))with the property that a Cartier divisorγonDextends to a Cartier divisor E onX if and only if

hγ,Ψi= 0 ∀Ψ∈H⁰(X,Ω²_X(D)).

The divisor E is unique up to rational equivalence.

Not surprisingly, we’ll construct such a pairing by using Grothendieck residues (see for instance [21] where the authors study the interplay between residues and zero-dimensional subschemes extension). WhenX is a toric surface we obtain an explicit formula forhγ,Ψi, generalizing a theorem of Wood [34]. To prove Theorem 1, we compute the cohomological obstruction to extend line bundles fromD toX and then we use the Dolbeault ¯∂-resolution and residue currents to make explicit the conditions.

Application to polynomial factorization. If two algebraic curves of fixed de- gree osculate each other on a finite subset with sufficiently big contact orders, they necessarily have a common component. This basic observation permits to derive from Theorem 1 a sketch of algorithm which computes the absolute factorizion of a bivariate polynomialf. The polynomialf is assumed to be defined over a subsfield K⊂Cand we look for its irreducible decomposition overC.

Under the assumptionf(0)6= 0, we can associate tof a smooth toric completion X ofC² whose boundary

∂X=D1+· · ·+Dr

is a normal crossing toric divisor and such that the curve C ⊂X of f does not contain any torus fixed points ofX. A Minkowski sum decomposition

Nf =P+Q

of the Newton polytope off corresponds to a line bundle decomposition OX(C)' LP⊗ LQ,

whereLP andLQ are both globally generated with at least one non trivial global section.

We obtain the following result

Theorem 2. There exists an unique effective divisor D with support |∂X| and rationally equivalent to C+∂X. Let γ be the restriction of C to D and suppose thatNf =P+Q. There exists an absolute factorqoff with Newton polytopeQif and only if there exists0≤γ⁰≤γ such that

deg(γ⁰·D_i) =degL_Q|Di, i= 1, . . . , r

and so that the osculation conditions hold for the pair (D, γ⁰). The factor q can be explicitely computed from γ⁰ by solving a sparse linear system of 2×V ol(Q) equations and Card(Q∩Z²)unknowns.

When the facet polynomials of f are square free over ¯K, we can derive from Theorem 2 the sketch of a vanishing-sum LR-algorithm (Subsection 3.4). It first computes the Newton polytope decomposition

N_f =Q₁+· · ·+Q_s

associated to the absolute decomposition of f. Then it computes the associated irreducible absolute factorization

f =q1× · · · ×qs

with floatting calculous and with a given precision. The numerical part of our al- gorithm reduces to the absolute factorization of the univariate exterior facet poly- nomials and the algorithmic complexity depends now on the Newton polytopeNf

instead of the usual degreed=deg(f). In particular, we fully profit from the com- binatoric restrictions imposed by Ostrowski’s conditionsNpq=Np+Nq (see [28]).

This permits to reduce the number of possible recombinations when compared to the Galligo-Rupprecht algorithm [13].

Finally, let us mention that Theorem 1 concerns also non toricC²-completions.

In theory, this permits to exploit the information given by thenon toricsingularities of C along the boundary of X when f has non reduced facet polynomials (see Subsection 3.6).

A formal study of algorithmic complexity, as well as questions of using non toric singularities information will be explored in a further work.

Related results. Our method is inspired by an algorithm presented in [12] that uses generical toric interpolation criterions [32] in an open neighborhood of ∂X (see Subsection 3.5). By using combinatorics tools, the authors in [1] obtain a comparable Hensel lifting process which there too takes in account the geometry of the Newton polytope. Finally, let us mention [3], where the authors reduce the factorization of a sparse polynomial to smaller dense factorizations.

Organization. The article is organized as follow. Section 2 is devoted to Algebraic Osculation. We introduce the problem in Subsection 2.1 and we construct the residue pairing in Subsection 2.2. We enounce precisely Theorem 1 in Subsection 2.3 and we give the proof in Subsection 2.4. We give an explicit formula for the osculation criterions whenX is a toric surface in Subsection 2.5. In Section 3, we pay attention to polynomial factorization. We prove Theorem 2 in Subsections 3.2 and 3.3 and we develop the sketch of a sparse polynomial factorization algorithm

in Subsection 3.4. We compare the underlying algortihmic complexity with related results in Subsections 3.5 and discuss non toric information in Subsection 3.6. We conclude in the last Subsection 3.7.

Aknowledgment. We would like to thanks Michel Brion, St´ephane Druel, Jos´e Ignacio Burgos and Martin Sombra for their disponibility and helpfull comments.

We thanks Mohamed Elkadi and Andr´e Galligo who suggested me to pay attention to sparse polynomial factorization.

2. Algebraic osculation

We prove here our main result giving criterions for algebraic osculation on the boundary of a smooth projective completion of the complex plane. In Subsection 2.5, we make explicit the osculation criterions in the toric case.

2.1. Notations and motivation. In all the sequel, (X,OX) designs a smooth projective surface where

X =X₀t |∂X|

is the disjoint union of an affine surface X₀ 'C² and of the support of a simple normal crossing divisor

∂X=D1+· · ·+Dr.

We say thatX is a completion of the complex plane with boundary ∂X.

Anosculation data on the boundary ofX is a pair (D, γ) where D= (k₁+ 1)D₁+· · ·+ (k_r+ 1)D_r

is an effective divisor with support |D| =|∂X| and γ is a Cartier divisor on the subscheme (|D|,O_D). By abuse of language, we’ll writeD= (|D|,O_D).

Anosculating divisor for (D, γ) is a Cartier divisorE onX which restricts toγ onD. That is

i^∗(E) =γ,

where i : D → X is the inclusion map. In other words, we are looking for a divisorE with prescripted restriction to the k_i^th-infinitesimal neighborhood ofDi. In general, such an osculating divisor does not exist and we are interested here to determine the necessary extra conditions.

We say thatγ has support |Γ|, where Γ designs the zero-cycleγ·∂X. Thus,γ can be uniquely written as a finite sum

γ= X

p∈|Γ|

γp,

eachγp being the restriction toD of a germ of an analyticcycle ofX γep=div(fp)

atp. Ifpbelongs to the smooth part of the boundary andeγpis irreducible, a curve restricts to γp at p if and only if it has contact order at least kp with eγp, where k_p+ 1 is the multiplicity ofDatp. This observation motivates the terminology of algebraic osculation.

2.2. Residues. It’s a classical fact that Grothendieck residues play a crucial role in osculation and interpolation problems. Let us mention for instance [19], [23], [31] and [32] for interpolation results and [21] for the interplay between residues and subscheme extensions. Not surprisingly, residues will appear here too.

Let Ω²_X be the canonical bundle ofX. We identify the line bundle Ω²_X(D) with the sheaf of meromorphic forms with polar locus bounded byD. Thus, any global section Ψ of Ω²_X(D) restricts to a closed 2-form on X₀. Since X₀ 'C² is simply connected, there exists a rational 1-formψ onX such that

dψ_|X0 = Ψ_|X0.

Forp∈ |Γ|, we denote byψp the germ ofψin the chosen local coordinates. Let hp be a local equation of D at p. Thus hpψp is holomorphic at p. Suppose for a while that f_p is holomorphic and irreducible. Then, following [20], we define the Grothendieck residue atpof the germ of meromorphic two formdf_p∧ψ_p/f_p as

res_phdf_p fp

∧ψ_pi := lim

→0

1 (2iπ)²

Z

u_p()

df_p fp

∧ψ_p, (1)

whereu_p() ={xclose top, |fp(x)|=₁,|hp|=₂}.

This definition does not depend on the choice of local coordinates [20]. By Stokes Theorem, it only depends on dψ= Ψ. Moreover, the local duality Theorem [20]

implies that (1) only depends on f_p modulo (h_p). That is, (1) depends on γ_p and not on the chosen lifting eγ_p = {f_p = 0}. By linearity, we can extend (1) to any germ of analytic cycleeγp=div(fp). Finally, we deduce that (1) defines a bilineary operator

hγ,Ψip:=res_phdf_p fp

∧ψ_pi

between the group of Cartier divisor of D and theC-vector spaceH⁰(X,Ω²_X(D)).

We refer to the proof of Theorem 1 (Subsection 2.4) for more details.

2.3. Criterions for algebraic osculation. We keep previous notations. Our main result is the following

Theorem 1. Let (D, γ) be an osculating data on the boundary ofX.

1. There exists a Cartier divisor E onX which restricts to γ if and only if X

p∈|Γ|

hγ,Ψi_p= 0 f or all Ψ∈H⁰(X,Ω²_X(D)).

(2)

The divisor E is unique up to rational equivalence.

2. If moreoverγ is effectif and

H¹(X,OX(E−D)) = 0,

then there exists an effective divisor ofX which restricts to γ onD.

The necessity of (2) follows from the Residue Theorem [20]. The difficult part consists to show sufficiency of conditions. The proof will be given in the next Subsection 2.4.

Let us first illustrate Theorem 1 on a simple example.

Example 1 (the Reiss relation). LetX =P²and consider a finite collection ofd >0 smooth analytic germs eγp transversal to a lineL ⊂P². Suppose that we look for an algebraic curveC ⊂P² of degree d which osculates each germ with a contact order≥2. This problem leads to the osculation data (D, γ), where D= 3Landγ is the restriction toDofP

p∈|Γ|eγ_p. There is an isomorphism H⁰(P²,Ω²_P2(3L))'H⁰(P²,O_P²)'C

and we can check that the form Ψ whose restriction to C² = P²\L is given by Ψ_|C2 = dx₀∧dy₀ is a generator. Thus we can choose ψ_|C2 = x₀dy₀. Letting x₀=X/Z andy₀=Y /Z we obtain

ψ=X(ZdY −Y dZ) Z³

in theP² homogeneous coordinates [X :Y :Z]. Up to an affine change of coordi- nates, we can suppose thatγ is supported in the affine chartY 6= 0. In the new affine coordinatesx=Z/Y andy=X/Y, the lineLhas local equationx= 0 and ψ=ydx/x³. Moreover, we can choose a Weierstrass equation

eγp=y−φp(x)

for the smooth germeγ_p, where φ_p∈C{x}. By Cauchy formula, we obtain resp

hydx∧d(y−φp) x³(y−φ_p)

i

=res0

h

φp(x)dx x³ i

=1 2φ⁰⁰_p(0),

where res0 is an univariate residue andφ⁰⁰_p is the second derivative ofφp. Finally, (2) is here equivalent to that

X

p∈|Γ|

φ⁰⁰_p(0) = 0.

(3)

For degree reasons, any osculating divisor is rationally equivalent to dL and by Serre duality

H¹(O_P2(d−3))'H⁰(O_P2(−d)) = 0.

Thus (3) is the unique relation necessary for that there exists an osculating curve C for (D, γ).

It’s easy to see directly necessity of (3). An osculating curve is given by a degree dpolynomialC={f(x, y) = 0} that can be factorized

f(x, y) = Y

p∈|Γ|

(y−up(x)) in C{x}[y]. Since deg(f) =d, the sum P

p∈|Γ|up(x) is a degree 1 polynomial and the relation

X

p∈|Γ|

u⁰⁰_p(0) = 0

holds. IfCosculateseγp with contact oder 2, thenφp andup have the same Taylor expansion up to order 2, which directly shows necessity of (3). When we express the second derivative ofu_pin terms of the partial derivative off, we recover the classical Reiss relation [18]. This result is obtained by Griffiths-Harris in [20], Chapter 6.

2.4. Proof of Theorem1. The Cartier divisorγonDcorresponds to a line bundle L ∈P ic(D) over D together with a global meromorphic sectionf. An osculating divisor for (D, γ) corresponds to a line bundleL ∈e P ic(X) together with a global meromorphic sectionfewhich restrict respectively to Landf onD.

The following lemma gives the formal cohomological obstruction for the exten- sion of L. All sheaves are considered here as analytic sheaves and any sheaf on a subschemeY ⊂X is implicitly considered as a sheaf on X by zero extension.

Lemma 1. There is a decomposition ofP ic(D)in a direct sum P ic(D) =P ic(X)⊕H¹(D,OD).

Proof. The classical exponential exact sequence exists for any curve ofX (reduced or not, see [5] p.63). We obtain the commutative diagram

(4)

0 −→ ZX −→ OX

exp(2iπ·)

−→ O^∗_X −→ 0

↓ ↓ ↓

0 −→ ZD −→ OD −→ O_D^∗ −→ 0

where ZD ⊂ OD is the subsheaf of Z-valued fonctions (so that ZD ' Z|D|) and vertical arrows are surjective restriction maps. SinceX is rational,Hⁱ(X,OX) = 0 fori >0. Using the associated long exact cohomological sequences, we obtain the commutative diagram

0 → P ic(X) ^δ→^X H²(X,ZX) → 0

↓ ↓r ↓r⁰ ↓

H¹(D,ZD) → H¹(D,OD) →^e P ic(D) ^δ→^D H²(D,ZD) → H²(D,OD).

Herer,r⁰ are restriction maps,δD,δX are the coboundary maps corresponding to Chern classes onDandX(see [5], Ch. 1 for the non reduced case) andeis induced by the exponential map.

We claim that r⁰ is an isomorphism. We have the exact sequence of constant sheaves

(5) 0→j_!(ZX0)→ZX →Z|D|→0

whereX0=X\|D|andj:X0→Xis the inclusion, giving the exact cohomological sequence

H_c²(X₀)→H²(X,ZX)→H²(|D|,Z|D|)→H_c³(X₀)

whereH_c^∗(X₀) is the compact support cohomology ofX₀. It is dual to the singular homology H_∗(X₀) of X₀ which vanishes in degree 1,2,3 since X₀ 'C², and the claim follows. Thus

δD◦r=r⁰◦δX

is an isomorphism, andP ic(X) is a direct summand ofP ic(D). There remains to show thatker(δD)'H¹(D,OD). The exponential cohomological sequence forX is exact in degree 0 so thatH¹(X,ZX)→H¹(X,OX) is injective andH¹(X,ZX) = 0.

We have just seen thatH_c²(X0) = 0, and finally (5) impliesH¹(|D|,Z|D|) = 0.

Corollary 1. There is an isomorphismP ic(X)'P ic(∂X).

Proof. By previous lemma, it’s enough to show that H¹(Dred,OD_red) = 0 where

∂X =Dred is the reduced part of D. We prove this by induction on the number of irreducible branches of Dred. SinceH¹(Z,ZZ) = 0 for a finite setZ, there is a surjection

H¹(|D|,Z|D|)→

r

M

i=1

H¹(Di,ZD_i).

Thus H¹(D_i,ZDi) = 0 for all i and each D_i is a rational curve. In particular, H¹(O_Di) = 0. Moreover, H_c¹(X₀) = 0 so that H⁰(X,ZX) → H⁰(|D|,Z|D|) is surjective and |D| is connected. This shows that |D| is a connected and simply connected tree of rational curves. There thus exists i so that Dred = Di +D⁰, where|Di| ∩ |D⁰|is a point andD⁰remains a connected and simply connected tree.

The short exact sequence

0→ OD_red → OD_i⊕ OD⁰ → O_Di·D⁰ →0 gives the long exact sequence in cohomology

0 → H⁰(OD_red)→H⁰(OD_i)⊕H⁰(OD⁰)→H⁰(OD_i·D⁰)

→ H¹(OD_red)→^r H¹(OD_i)⊕H¹(D⁰,OD⁰)→H¹(OD_i·D⁰).

By assumption,Dredis anormal crossing divisor andDi·D⁰={pt} as areduced subscheme. The diagram then begins by 0→C→C⊕C→C, which forcesrto be injective. SinceH¹(ODi) =H¹(O_{pt}) = 0, this givesH¹(OD_red)'H¹(OD⁰).

By previous corollary, there exists an unique line bundle L ∈e P ic(X) so that Le_|∂X' L_|∂X andL lifts toX if and only if

L0:=L ⊗Le⁻¹_|D ' OD.

Since δD(L0) = 0 (by construction), there existsβ ∈H¹(D,OD) withe(β) = L0. Moreovereis injective and

L₀' O_D ⇐⇒ β = 0.

There remains to make explicit such a condition. We first use Cech cohomology.

Suppose that L0 is given by the one cocycle class g ={gU V} ∈ H¹(D,O_D^∗), rel- ative to some open covering U of X. We can supposeU fine enough to ensure a logarithmic determination

ehU V := 1

2iπlog(egU V)∈ OX(U∩V),

whereeg_{U V} ∈ O^∗_X(U∩V) is some local lifting ofg_{U V}. Since δ_D(g) = 0, the classes hU V ∈ OD(U∩V) define a 1-cocycle class{hU V} ∈H¹(D,OD) which representsβ (this definition does not depend on the liftings). SinceX is rational, the structural sequence forD

0−→ O_X(−D)−→ O_X−→ O_D−→0

gives rise to a coboundaryisomorphism δ:H¹(D,OD)→H²(X,OX(−D)). Serre Duality gives a non degenerate pairing

H²(X,OX(−D))⊗H⁰(X,Ω²_X(D))−→^(·,·) H²(X,Ω²_X)^{T r}'C

where T r is the trace map [5]. We identify OX(−D) with the sheaf of functions vanishing onDand Ω²_X(D) with the sheaf of meromorphic 2-forms with polar locus

bounded byD. The Serre pairing (δ(β),Ψ) with Ψ∈H⁰(X,Ω²_X(D)) is represented by the 2-cocyle class

ζΨ:=n

(ehU V +ehV W+ehW U)Ψ_|U∩V∩Wo

∈H²(X,Ω²_X) andβ= 0 if and only if

T r(ζ_Ψ) = 0 ∀Ψ∈H⁰(X,Ω²_X(D)).

To make explicit the complex numbersT r(ζΨ), we use the Dolbeault ¯∂-resolution of Ω²_X. We denote byD^(p,q)_X the sheaf of germs of (p, q)-currents onX. Letψbe a germ of meromorphicq-form atp∈X. We recall for convenience that theprincipal value current [ψ] ∈ D_X,p^(q,0) and theresidue current ∂[ψ]¯ ∈ D_X,p^(q,1) have the Cauchy integral representations

h[ψ], θi:= lim

→0

1 2iπ

Z

U∩{|u|>}

ψ∧θ and

h∂[ψ], θi¯ := lim

→0

1 2iπ

Z

U∩{|u|=}

ψ∧θ,

whereθ is some form-test with appropriate bidegree andu= 0 is a local equation for the polar divisor of ψ in a small neighborhood U of p (see [30] for instance).

The Dolbeault resolution of Ω²_X is given by the exact complex of sheaves 0−→Ω²_X−→ D^[·] _X^(2,0)−→ D^{∂¯ (2,1)}_X −→ D^{∂¯ (2,2)}_X −→0.

This sequence breaks in the two short exact sequences

0→Ω²_X→ D^(2,0)_X → Z^∂¯ _X^(2,1)→0 and 0→ Z_X^(2,1)→ D_X^(2,1)→ D^{∂¯ (2,2)}_X →0, where Z_X^(2,1)⊂ D_X^(2,1) is the subsheaf of ¯∂-closed (2,1)-currents. Since the sheaves D^(p,q)_X are fine, we obtain the two coboundaryisomorphisms

H⁰(X,D^(2,2)_X )

∂H¯ ⁰(X,D^(2,1)_X )

δ₁

−→H¹(X,Z_X^(2,1)) and H¹(X,Z_X^(2,1))−→^δ2 H²(X,Ω²_X).

Thus, for any Ψ, there exists a global (2,2)-currentTΨ∈H⁰(X,D_X^(2,2)) whose class [T_Ψ] modulo ¯∂ is unique solution to ζ_Ψ=δ₂◦δ₁([T_Ψ]), and we have equality

T r(ζΨ) =hTΨ,1i.

To make explicitTΨ, we need the following lemma.

Lemma 2. 1. There exists an open tubular neighborhoodB of |D| and a Cartier divisoreγ onB which restricts toγ on D.

2. There exists is isomorphism L ' Oe X(E) for a Cartier divisor E which restricts to Γon Dred.

Proof. 1. By mean of partition of unity, we can construct ua C^∞ function onX which vanishes exactly on|D|, giving an open tubular neighborhood ofD

B={|u|< }.

Letf ={fU} be a meromorphic section ofL with Cartier divisorγ, and consider some local liftings fe_U ∈ MX(U), with MX the sheaf of meromorphic functions onB. We can choose V a sufficiently fine covering ofB in order to suppose that

U∩V ∩ |Γ|=∅for distinctU, V ∈ V, with moreoverfeU ∈ OX(U)^∗ forU∩ |Γ|=∅.

If now U intersects |Γ|, we can choose ⁰ < small enough so that feU ∈ MX(U) has nor poles nor zeroes on U ∩V ∩B⁰. In such a way, feU/feV is invertible on U∩V ∩B⁰, giving a Cartier divisorγeonB⁰ which restricts toγ onD.

2. LetF =OX(F) be some very ample line bundle on X with F an effective divisor which intersects properly|Dred|. The isomorphisms

OD_red(Γ)' L_|Dred'Le_|Dred

combined with the structural sequence forDredgives the exact sequence 0→L ⊗ Oe _X(nF−D_red)→L ⊗ Oe _X(nF)→ O_Dred(Γ +nΓ_F)→0 for any integern, where ΓF =F·Dred. Fornbig enough, we have

H¹(L ⊗ Oe X(nF−Dred)) = 0

by Serre Vanishing Theorem so that Γ +nΓ_F lifts to some Cartier divisor G on X. Then, the Cartier divisorE=G−nF restricts to Γ onD_red. By construction OX(E) and L=OD(γ) have the same restriction to ∂X =Dred, and Corollary 2

impliesL ' Oe X(E).

LetB,eγandEas in Lemma 2, withV the associated open covering ofB. Since OB(eγ)⊗ OB(−E) restricts toL0onD, we can choose the liftingsegU V =mU/mV, wherem={mU}_U∈V is the global meromorphic section of OB(eγ)⊗ OB(−E) with Cartier divisor

div(m) =eγ−E_|B

onB. By construction, m_U|U∩V ∈ O^∗_B(U∩V) for distinctU, V ∈ V. Thus, there exists a logarithmic determination

log(m_U|U∩V) :=

∞

X

n=1

1

n (1−mU|U∩V)n

of m_U|U∩V. Since div(m)·Dred = 0, the restriction of m to Dred is a constant, that we can suppose to be 1. There thus existsn₀∈Nsuch that the meromorphic function (1−m_U)ⁿ vanishes on the divisorD∩U for alln≥n₀and allU ∈ V.

Let Ψ∈H⁰(X,Ω²_X(D)). By the Duality Theorem [30], we have 1−mU

n∂¯ ΨU

= 0 ∀n≥n0

so that the principal value currents SU :=

n₀

X

n=1

1 n h

1−mU

ni

satisfy

(S_U−S_V) ¯∂ Ψ_U∩V

=

log(eg_U∩V)∂¯ Ψ_U∩V (6)

for allU, V ∈ V. We obtain in such a way a global (2,2)-current TB ∈Γ(B,D_B^(2,2)) onB, locally given by

TU = ¯∂SU∧∂[Ψ¯ U]

for all U ∈ V. Since ¯∂[Ψ_U] = 0 for all U with U∩ |D|=∅, we can extend T_B by zero to a global current T on X. Using (6) and the definition of the coboundary maps, it’s now straightforward to check thatT is solution to

δ₂◦δ₁(T) =ζ_Ψ.

Let nowψ be some rational 1-form onX so thatdψ_|X0 = Ψ_|X0. For allU ∈ V, we define the local currents

R_U :=

dS_U−hdm_U mU

i∧∂¯ ψ_U

and T_U⁰ = ¯∂hdm_U mU

i∧∂¯ ψ_U

.

From (6) we deduce equalities R_U|U∩V = RV|U∩V, and the RU’s define a global (2,1)-current RB on B. By assumption, mU/mV is invertible on U ∩V so that T_U|U∩V⁰ = T_V⁰_|U∩V, giving a global (2,2)-current T_B⁰ on B. Both currents are supported on|D| ⊂B and can be extended by zero to global currentsRandT⁰ on X.

SinceTU has support a finite set, Stokes Theorem gives equalities hTU,1i = h∂S¯ _U∧∂[Ψ¯ _U],1i

= h∂S¯ U∧∂d[ψ¯ U],1i

= h∂(dS¯ _U)∧∂[ψ¯ _U],1i=hT_U⁰ + ¯∂R_U,1i.

Thus, hT,1i=hT⁰+ ¯∂R,1i=hT⁰,1i. Sincediv(m) =eγ−E_|B onB, the Lelong- Poincar´e equation gives equality

hT⁰,1i=hT_B⁰ ,1i=h[γ]e ∧∂[ψ]¯ _|B,1i − h[E]_|B∧∂[ψ]¯ _|B,1i,

where [·] designs here the integration current associated to analytic cycles. The cur- rent [E]_|B∧∂[ψ]¯ _|Bis supported on the compact subset|Γ| ⊂B, and the integration current is ¯∂-closed. We deduce

h[E]_|B∧∂[ψ]¯ _|B,1i=h[E]∧∂[ψ],¯ 1i=h∂([E]¯ ∧[ψ]),1i= 0.

This is also equivalent to the Residue Theorem [20]. ForB small enough,B∩γeis a disjoint union of analytic cycleseγp={fp= 0}. Ifψpis the germ ofψatpin the chosen local coordinates, we obtain finally

T r(ζΨ) = X

p∈|Γ|

h[eγp]∧∂[ψ¯ p],1i

= X

p∈|Γ|

resp

dfp

fp

∧ψp

= X

p∈|Γ|

hγ,Ψip. This expression only depends onOD(γ) and Ψ by construction.

Conditions (2) of Theorem 2 are thus equivalent to thatL=OD(γ) extends to L ∈e P ic(X). IfLextends toL, we can use Vanishing Serre Theorem as in Lemmae 2 (with (D, γ) instead of (Dred,Γ)) and show that L ' Oe X(E) for some Cartier divisorEonX which restricts toγonD. The lifting bundleLebeing unique (up to isomorphism), the osculating divisorE is unique up to rational equivalence. This ends the proof of the first point.

If conditions (2) hold, thenL ' O_D(E) for some Cartier divisorE on X. By tensoring the structural sequence ofD withLe=OX(E), we obtain the short exact sequence

0→ OX(E−D)→ OX(E)→ OD(E)→0.

Ifγ is effectif and H¹(X,O_X(E−D)) = 0, the global sectionf ∈H⁰(D,O_D(E)) with zero divisorγ automatically lifts to fe∈ H⁰(X,O_X(E)) and C :=div₀(fe) is aneffective osculating divisor for (D, γ). This ends the proof of Theorem 1.

2.5. An explicit formula in the toric case. We show now that we can make explicit the conditions (2) in the case of a toric varietyX. We refer the reader to [14] and [11] for an introduction to toric geometry.

2.5.1. Preleminaries. Suppose thatX is atoric surface containingX₀ as an union of open orbits. Then,X has toric divisorsD₀, . . . , D_r+1 where

∂X=D1+· · ·+Dr

is the boundary ofXandD₀andD_r+1are the Zariski closure of the one dimensional open orbits ofX₀'C².

Let Σ be the fan ofX. We denote byρ_i∈Σ the ray associated to the toric divisor D_i. Since Σ is regular, we can order theD_i’s in such a way that the generators η_i of the monoidsρ_i∩Z² satisfy

det(ηi, ηi+1) = 1 (with conventionη_r+2=η₀).

We denote byU_i the affine toric chart associated to the two-dimensional cone ρ_iR⁺⊕ρ_i+1R⁺. Thus,

Ui=SpecC[xi, yi]'C²,

where torus coordinates t = (t1, t2) and affine coordinates (xi, yi) are related by relations

t^m=x^hm,η_i ⁱⁱy^hm,η_i ⁱ⁺¹ⁱ

for allm= (m1, m2)∈Z², wheret^m=t^m₁¹t^m₂² (see [11]). With these conventions, the toric divisorsDi andDi+1 have respective affine equations

D_i|Ui={xi= 0} and D_i+1|Ui ={yi= 0}

and|Dj| ∩Ui=∅forj6=i, i+ 1.

We can associate to any toric divisorE=Pr+1

i=0e_iD_i a polytope PE={m∈R²,hm, ηii+ei≥0, i= 0, . . . , r+ 1},

where h·,·i designs the usual scalar product in R². Each Laurent monomial t^m defines a rational function onX with divisor

div(t^m) =

r+1

X

i=0

hm, ηiiDi

and there is a natural isomorphism

H⁰(X,OX(E))' M

m∈PE∩Z²

C · t^m, (7)

whereH⁰(X,OX(E)) is the vector space of rational functionsh∈C(X) for which div(h) +E≥0. Finally, we recall that the toric divisor

−K_X:=D₀+· · ·+D_r+1 is an anticanonical divisor forX.

2.5.2. Explicit osculating criterions. Let (D, γ) be an osculation data on∂X. We note Γ =γ·∂X and Γi=γ·Di. We assume that the following hypothesis holds:

(H₁) The zero-cycleΓ is reduced and does not contain any torus fixed points.

Thus, the germseγp are smooth, irreducible, and have transversal intersection with

∂X. For each p∈ |Γ|, there is an unique component Di containingpand we can choose the Weiertrass equation

f_p(x_i, y_i) =y_i−φ_p(x_i)

for the associated germeγp. The analytic functionφp∈C{xi}is well-defined modulo (x^k_iⁱ⁺¹) and does not vanish at 0. We obtain the following

Corollary 2. With previous hypothesis and notations, conditions(2) become

X

i∈Im⁻

1 (−hm, ηii)!

"

∂^−hm,ηii

∂x^−hm,η_i ⁱⁱ X

p∈|Γi|

φ^hm,ηp ⁱ⁺¹ⁱ

hm, ηi+1i

!#

(0) = 0 (8)

for allm∈PD+KX∩Z², where

I_m⁻={i∈ {1, . . . , r}, hm, ηii ≤0}, and with convention φ^k_p/k:=log(φp)fork= 0.

The previous expression is well-defined, being a rational expression on the suc- cessive derivatives and logarithmic derivatives of theφ_p’s evaluated at 0 (recall that φ_p(0)6= 0). Note that we don’t exclude that Γ_i= 0 for somei.

Example 2 (Wood’s theorem). We keep the same hypothesis and notations of Ex- ample 1, with now an osculating orderk≥2. ThusD= (k+ 1)L. Hypothesis (H1) holds and by Corollary 1, there exists an osculating divisor for (D, γ) if and only if

∂^m1+m2

∂x^m1+m2 X

p∈|Γ|

φ^m_p¹ m₁

(0) = 0, ∀m∈(N^∗)², m1+m2≤k.

(9)

Note thatH¹(O_P2(d−k−1)) = 0 ifk≤d+ 1. In such a case, (9) is also sufficient for the existence of an osculating algebraic curve. We recover here a theorem of Wood, [34].

2.5.3. Proof of Corollary 2. Form∈Z², we denote by Ψmthe rational form given by

Ψ_m|(C^∗₎2 =t^mdt1∧dt2

t₁t₂

in torus coordinates. Then,div(Ψm) =div(t^m)−KX and there is equality H⁰(X,Ω²_X(D)) = M

m∈PD+KX∩Z²

C · Ψm.

Letm∈P_D+KX. In particularhm, η0i ≥1 and we can suppose thatm₂6= 0. The form Ψ_m is holomorphic in X₀ and Ψ_m|X0 =d(ψ_m|X0) where ψ_m is the rational

1-form onX determined by its restriction ψ_m|(C∗)² = t^m

m2

dt₁ t1

to the torus. Letp∈ |Γi|. We compute the associated residue in the chartUi. If we let (e1, e2) be the canonical basis of R², we obtain

ψ_m|Ui =x^hm,η_i ⁱⁱy^hm,η_i ⁱ⁺¹ⁱ 1 m₂

he1, ηiidxi

x_i +he1, ηi+1idyi

y_i .

If hm, ηii > 0, the form ψm is holomorphic at p ∈ |Γi| and resp(^df_f^p

p ∧ψm) = 0.

Suppose now thathm, ηii ≤0. The Cauchy integral representation for Grothendieck residues of meromorphic forms does not depend onfor_i small enough, thanks to Stokes Theorem. Thus, for sufficiently small_i’s, we obtain

resp

h

x^hm,η_i ⁱⁱy_i^hm,ηi+1idfp

f_p ∧dyi

y_i i

= − 1

(2iπ)² Z

|xi|=1,|yi−φp|=2

x^hm,η_i ⁱⁱy_i^hm,ηi+1i dφp

y_i−φ_p ∧ dyi

y_i

= 1

(2iπ)² Z

|xi|=1

Z

|yi−φp|=2

y_i^{hm,ηi+1i−1} dyi

yi−φp

!

x^hm,η_i ⁱⁱdφp

= 1

2iπ Z

|xi|=

x^hm,η_i ⁱⁱφ^hm,η_p ⁱ⁺¹ⁱ⁻¹φ⁰_pdx

= 1

(−hm, η_ii −1)!

"

∂^−hm,ηii

∂x^−hm,η_i ⁱⁱ

φ^hm,ηp ⁱ⁺¹ⁱ

hm, η_i+1i

!#

xi=0

where the two last equalities are application of the Cauchy formula. Ifhm, ηi+1i= 0, φ^hm,ηp ⁱ⁺¹ⁱ

hm, ηi+1i :=log(φp)

is only defined up to some constant (recall thatφ_p(0)6= 0) but 0∈/P_D+KX so that hm, η_ii>0 in that case. Thus previous expression only depends on the logarithmic derivatives ofφp, so is well-defined. In the same way, but simpler, we find

resp

h

x^hm,η_i ⁱⁱy_i^hm,ηi+1idfp

f_p ∧dxi

x_i i

= −1

(−hm, η_ii)!

"

∂^−hm,ηii

∂x^−hm,η_i ⁱⁱ

φ^hm,η_p ⁱ⁺¹ⁱ

#

x_i=0

,

the minus sign coming from the chosen ordering of the numerator’s factors. We deduce

resp

dfp

f_p ∧ψm

= C

m₂× 1 (−hm, ηii)!

"

∂^−hm,ηii

∂x^−hm,η_i ⁱⁱ

φ^hm,ηp ⁱ⁺¹ⁱ

hm, ηi+1i

!#

xi=0

where

C=he1, ηiihm, ηi+1i − he1, ηi+1ihm, ηii=m2det(ηi, ηi+1) =m2.

This ends the proof of Corollary 1.

If two algebraic curves of fixed degree osculate each other on a finite subset with sufficiently big contact orders, they necessarily have a common component.

We show in the next section that this basic fact permits to apply Theorem 1 and Corollary 1 to the Computer Algebra problem of polynomial factorization.

3. Application to polynomial factorization.

This section is devoted to develop a new algorithm to compute the absolute factorization of a bivariate polynomialf. The method we introduce uses vanishing- sums in the vain of the Galligo-Rupprecht algorithm [13]. The main difference is that the underlying algorithmic complexity depends now on the Newton polytope Nf instead of the usual degreed=deg(f). Our algorithm is comparable to a toric version of the Hensel lifting, in the spirit of the Abu Salem-Gao-Lauder algorithm [1]. We prove our main results Theorem 2 and Proposition 1 in Subsections 3.2 and 3.3. We describe a sketch of algorithm in Subsection 3.4 and we comment and compare it with related results in Subsection 3.5. We discuss non toric singularities in Subsection 3.6 and we conclude in the last Subsection 3.7.

3.1. Preliminaries and notations. Letf ∈K[t1, t2] be a bivariate polynomial with coefficients in some subfieldK⊂C. By absolute factorization off, we mean its irreducible decomposition in the ringC[t1, t2], computed with floatting numbers with a given precision.

Suppose thatf has the monomial expansion f(t) = X

m∈N²

c_mt^m,

with the usual notations m= (m₁, m₂)∈Z² andt^m=t^m₁¹t^m₂². We denote byN_f the Newton polytope off, convex hull of the exponentsm∈N² for whichc_m6= 0.

We recall that by a theorem of Ostrovski [28], we have relations Nf₁f₂ =Nf₁+Nf₂

for any polynomialsf₁, f₂, where + designs here the Minkowski sum.

We definitively assume that the following hypothesis holds.

(H2) The Newton polytope of f contains the origin.

This equivalent to thatf(0,0) 6= 0. Anexterior facet of N_f is a one-dimensional face F of N_f whose normal inward primitive vector has non simultaneously posi- tive coordinates. The associated normal ray ofNf is called anexterior ray. The associated exterior facet polynomialfF off is defined to be

fF = X

m∈F∩Z²

cmt^m.

The facet polynomials have a one dimensional Newton polytope and become uni- variate polynomials after a monomial change of coordinates.

We construct now a toric completion of C² associated to the exterior facets of N_f. To this aim, we consider thecomplete simplicial fan Σ_f ofR² whose rays are

Σf(1) ={exterior rays of Nf} ∪ {(0,1)R⁺,(1,0)R⁺}.

The toric surfaceXf =XΣ_f is a simplicial toric completion ofC²=SpecC[t1, t2] whose boundary∂X_f =X_f\C²is the sum of the irreducible toric divisors associated to the exterior rays ofN_f.

By hypothesis (H2), the fan Σf refines the normal fan ofNf. The intersection of the Zariski closureCf ⊂Xf of the affine curve {f = 0} ⊂C² with the irreducible components of the boundary∂Xfcorresponds to thenon trivialroots of the exterior facet polynomial of f (see [24] for instance). In particular, the curveC_f does not contain any torus fixed points. We say thatfsatisfies hypothesis (H₁) of Subsection 2.5 when the zero-cycle Γ_f =C_f·∂X_f does. This corresponds to the case of exterior facet polynomials off with a square free absolute decomposition inC[t, t⁻¹].

The boundary of X_f contains the singular locus S := Sing(X_f) of X_f and is generally not a normal crossing divisor. To this aim, we consider a toric desingu- larization

X→^π Xf

ofXf, whose exceptional divisor E satisfiesS =π(|E|). The smooth surfaceX is now a projective toricC²-completion with a toric normal crossing boundary

∂X=D1+· · ·+Dr

whose support contains |E|. SinceCf∩S =∅, the total transform C =π⁻¹(Cf) of Cf has a proper intersection with ∂X and the irreducible decomposition of C corresponds to the absolute factorization of f. Note that C is not necessarily reduced.

As in Subsection 2.5, we denote byρ0, . . . , ρr+1 the rays of the fan Σ ofX, and byη₀, . . . , η_r+1 their primitive vectors. Soη₀= (0,1) andη_r+1= (1,0). The curve C has a positive intersection C·D_i >0 whenρ_i ∈Σ(1) is an exterior ray ofN_f, andC·D_i= 0 whenρ_i∈Σ(1)\Σ_f(1) (that is whenπ(|D_i|)⊂S).

We’ll need the following lemma

Lemma 3. There is an unique effective divisor D linearly equivalent to C+∂X with support|∂X|. Moreover, the following equality

P_D+KX ∩Z²=N_f∩(N^∗)² holds.

Proof. The polynomialf extends to a rational function onX whose polar divisor div_∞(f) is supported by|∂X|. Thus, the divisor

D:=div_∞(f) +∂X

is effective, with support |∂X|, and rationally equivalent to C+∂X. If D⁰ is an other candidate, thenD−D⁰ =div(h) for someh∈C(X) with no poles nor zeroes outside|∂X|. Since X\ |∂X| 'C², the rational functionhis necessarily constant andD=D⁰.

Let us noteG :=div_∞(f) and PG its associated polytope. By (7), Nf is con- tained in PG with at least one point on each face of PG. Since the fan of X refines the nornal fan off, both polytopes have parallelic facets and it follows that PG=Nf.Now, D+KX =G+KX+∂X=G−D0−Dr+1 so that

PD+K_X ={m∈PG,hm, η0i ≥1, hm, ηr+1i ≥1}.

We finally obtain equalityPD+K_X ∩Z²=Nf∩(N^∗)².

3.2. Computing the Newton polytopes of the absolute factors. We show here how to recover the Newton polytopes of the irreducible absolute factors ofNf. We recall that the Newton polytope of a factor off is necessarily a Minkowski summand ofN_f. For a general polytopeQ, we denote byQ⁽ⁱ⁾the set ofm∈Qfor which the scalar producthm, ηiiis minimal.

We obtain the following

Theorem 2. 1. Let f ∈ K[t1, t2] which satisfies (H2). Let π, X and C be as before and denote by γ the restriction ofC to the divisorD of Lemma3. LetQbe a lattice Minkowski summand ofNf.

There exists an absolute factorqoff with Newton polytopeQif and only if there existsγ⁰ ≤γ an effective Cartier divisor onD with

deg(γ⁰·Di) =Card(Q⁽ⁱ⁾∩Z²)−1, i= 1, . . . , r (10)

and such that(2)holds for the osculation data(D, γ⁰)on the boundary ofX. These conditions are independant of the choice of π. The polynomial q can be recovered fromγ⁰.

Proof. A Minkowski sum decompositionN_f =P+Qcorresponds to a line bundle decompostion

OX(G) =OX(GP)⊗ OX(GQ),

whereG,GP andGQare the polar divisors of the rational functions determined by polynomials with respective polytopeNf,P andQ. Note thatG=GP+GQ. Since Σ refines the normal fan of each polytope, we have equalitiesNf =PG, Q=PG_Q

and P =PG_P (see Lemma 3) and all involved line bundles are globally generated overX (see [14]).

A factor of f with Newton polytope Q corresponds to a divisor C⁰ ≤C lying in the complete linear system |OX(GQ)|. It defines a Cartier divisor γ⁰ =C_|D⁰ ≤ C_|D=γ onD. Thus, conditions (2) hold for the osculation data (D, γ⁰) and

deg(γ⁰·Di) =deg(C⁰·Di) =deg(OX(GQ)_|Di)

for alli= 1, . . . , r. Since OX(GQ) is globally generated overX, toric intersection theory gives equalities

deg(OX(G_Q)_|Di) =Card(Q⁽ⁱ⁾∩Z²)−1

for alli = 1, . . . , r. This shows necessity of conditions. Note that i ∈ Iπ implies C⁰·D_i= 0 in which caseQ⁽ⁱ⁾is reduced to a point.

Suppose now that (D, γ⁰) satisfies (10) and (2) and letFbe an osculating divisor asociated to that data. We deduce equalities

degOX(F)_|Di =deg(γ⁰·D_i) =degOX(G_Q)_|Di for alli= 1, . . . , r. Thus, there is an isomorphism

O∂X(F)' O∂X(GQ)

andOX(F)' OX(GQ) by Corollary 1 of Subsection 2.4. Finally, (2) and (10) hold for (D, γ⁰) if and only if

OD(γ⁰)' OD(G_Q).

Let us show that there exists in such a case an effective osculating divisor. The problem is that H¹(X,OX(GQ −D)) does not necessarily vanish. To avoid this difficulty, we introduce the effective divisor

D_Q :=G_Q+∂X≤G+∂X=D.

ThenOD_Q(GQ)' OD_Q(γ⁰_|D

Q) and

H¹(X,OX(GQ−DQ))'H¹(X,OX(−∂X)).

Since∂X is reduced and connected, the restriction mapH⁰(OX)→H⁰(O∂X) is an isomorphism and we deduce equalityH¹(X,OX(−∂X)) = 0.By Theorem 1, there thus existsC⁰∈ |O_X(G_Q)|with

C_|D⁰

Q =γ_|D⁰

Q.

Such a curve has a proper intersection with the remaining toric divisors D0 and Dr+1 (otherwise the support ofγ⁰ would contain a torus fixed point). We deduce that there exists an unique Cartier divisorγ₀ onD+D₀+D_r+1 so that

γ_0|D=γ⁰ and γ_0|D0+Dr+1 =C⁰·D0+C⁰·Dr+1

We have equalityD+D0+Dr+1=G−KX and we obtain

H¹(X,OX(G_Q−D−D₀−D_r+1) ' H¹(X,OX(G_Q−G+K_X) ' H¹(X,OX(G−GQ))^ˇ ' H¹(X,OX(GP))^ˇ= 0.

The second isomorphism is Serre Duality, and the last equality comes from the fact that the H¹ of a globally generated line bundle on a toric surface vanishes (see [14]). Last equality combined with the short exact sequence

0→ OX(GQ−D−D0−Dr+1)→ OX(GQ)→ OD+D₀+D_r+1(γ0)→0, imply that there exists C⁰⁰ ∈ |OX(GQ)| which restricts to γ0 onD+D0+Dr+1. A fortiori, C⁰⁰ restricts to γ⁰ on D. Since GQ −DQ = −∂X < 0, we have H⁰(X,OX(GQ −DQ)) = 0. Thus C⁰ = C⁰⁰, the two curves having the same restriction toD_Q.

There remains to show that C⁰ ≤C. Suppose that there exists an irreducible component C₀ ofC⁰ with proper intersection withC. Then, we can compute the intersection multiplicity ofC₀ andC at anyp∈X. It is given by

multp(C0, C) =dim_C OX,p

(fp, gp)

wherefpandgp are respective local equations forC andC0 atp. By assumption, C_0|D≤C_|D

so that the class ofg_p inO_D,pdivides that off_p. We deduce that fp=upgp+vphp

for some holomorphic functionsu_p, v_p, whereh_p is a local equation for D. Thus, we obtain inequality

dim_C OX,p

(fp, gp) ≥dim_C OX,p

(gp, hp) =multp(C0, D),