Topology and Factorization of Polynomials

TOPOLOGY AND F A CTORIZA TION OF POL YNOMIALS HANI SHAKER Abstract. F or any polynomial P ∈ C [ X 1 , X 2 , ..., X n ], we describe a C -vector space F ( P ) of solutions o f a linear system of equatio ns coming from some alge- braic partial differential equations such t hat the dimension o f F ( P ) is the n um ber of irreducible factors of P . Mor eov er, the knowledge of F ( P ) gives a co mplete fac- torization of the p o lynomial P b y taking gcd’s. This gener alizes previous r esults by Rupp ert and Gao in the case n = 2 . 1. Introduction Let K b e the alg ebraic closure of a field k and let k [ X 1 , X 2 , ..., X n ] b e the p olyno- mial ring in n indeterminates. The zero se t of a p olynomial P ∈ k [ X 1 , X 2 , ..., X n ] of deg d > 0 is a h yp ersurface V ( P ) in K n . As the p olynomial ring is a factoria l ring, w e can write P = Q s i =1 P i , where P i are the irreducible factors of P in K [ X 1 , X 2 , ..., X n ] . W e assume that the factors P i are distinct, i.e. P is a r e duc e d p olynomial . The prime factorization of P corresp onds to the decomp osition in to irreducible comp o- nen ts V ( P ) = Q s i =1 V ( P i ) of the hy p ersurface V ( P ). A nat ural question to ask is: ” How c an we c omp ute s , the numb er of irr e ducible factors of P (r esp. irr e ducible c om p onents of V ( P ) ) fr o m the c o efficien ts of P ?” A v arian t of this problem (called the absolute factorization pr oblem ) is when P is assumed to be irreducible in k [ X 1 , X 2 , ..., X n ], see [1]. In this pap er w e recall in Section 2 briefly Gao ’s results in the case n = 2, see [3], and then some usual tec hniques for reducing t he case n > 2 to the case n = 2 b y taking generic linear sections , see [1]. Since all these reduction t echniq ues are not easy to use in practice (since the notion of a generi c linear sec tion is quite subtle as w e sho w b y some examples ), w e dev elop in Sections 3 a nd 4 of our note a direct approach to the case n > 2. Unlik e Rupp ert-Gao’s approac h, whic h is purely algebraic and w orks o ve r an y field k of c haracteristic zero or of relativ ely large c ha racteristic, our approac h is top ological, using de Rham cohomology , and hence w orks only for the algebraically closed subfields of the field of complex n um b ers C . 2000 Mathematics Subje ct Classific ation. Primary 12D05; Secondary 14F40,14J7 0 ;. Key wor ds and phr ases. polynomial ring, factorizatio n, de Rham cohomology . 1 2 HANI SH AKER 2. Rupper t-Gao’s idea and the reduction techniques Assume that n = 2 and denote b y X, Y the tw o indeterminates. If P ( X , Y ) = Q s i =1 P i ( X , Y ) , is the factorization of P in to irreducible factors in K [ X, Y ], then b y taking the pa r tial deriv ativ es on b oth side s, we ha ve (2.1) P X = s X i =1 ( Y j 6 = i P j ) ∂ P i ∂ X = s X i =1 g i where g i = ( Y j 6 = i P j ) ∂ P i ∂ X and also (2.2) P Y = s X i =1 ( Y j 6 = i P j ) ∂ P i ∂ Y = s X i =1 h i where h i = ( Y j 6 = i P j ) ∂ P i ∂ Y . Note that we can write ∂ ∂ X (log P i ) = 1 P i ∂ P i ∂ X = g i P , ∂ ∂ Y (log P i ) = 1 P i ∂ P i ∂ Y = h i P whic h yields (2.3) ∂ ∂ Y ( g i P ) = ∂ ∂ X ( h i P ) for i = 1 , ..., s. Definition 2.1. Let P ( X , Y ) ∈ K [ X , Y ] b e suc h that deg X ( P ) = m 1 , deg Y ( P ) = m 2 . Then the bidegree of P is defined as bideg( P ) = (deg X ( P ) , deg Y ( P )) = ( m 1 , m 2 ) . In our case, w e obv iously ha ve bideg( g i ) ≤ ( m 1 − 1 , m 2 ) and bideg ( h i ) ≤ ( m 1 , m 2 − 1) . Definition 2.2. Let F ( P ) b e the K − v ector space of solutions ( v , w ) ∈ K [ X , Y ] 2 of the partial differen tial equation ∂ ∂ Y ( v P ) = ∂ ∂ X ( w P ) suc h that bideg( v ) ≤ ( m 1 − 1 , m 2 ) , bideg( w ) ≤ ( m 1 , m 2 − 1) . This partial differen tial equation w as first considered b y Rupp ert [5], [6]. More- o v er, it w as clear to R upp ert and Ga o that this is just the condition that a certain 1-form is closed, see the commen t just b efore The orem 2.1 in [3]. Theorem 2.3. (Gao’s Theorem) [3] If P ( X , Y ) = Q s i =1 P i ( X , Y ) is the factorization of P int o irr e ducible factors in K [ X , Y ] , then s = dim K F ( P ) and the set { ( g i , h i ) | i = 1 , ..., s } is a b a s is fo r F ( P ) . TOPOLOGY AN D F A CTORIZA TION OF POL YNO MIALS 3 Corollary 2.4. (i) P is i rr e ducible if and only if dim K F ( P ) = 1 . (ii) P i = g.c.d . ( P, v − λ i P X ) wher e v = Σ s i =1 λ i g i is a generic ve ctor in the ve ctor sp ac e E ( P ) obtaine d fr o m F ( P ) by pr oje cting on the first fa c tor. Here λ = ( λ 1 , ..., λ s ) ∈ C s and the genericit y means that λ has to a v oid a pro p er Zariski closed subset of C s . The first claim is an ob vious consequence of Theorem 2.3 and w as obtained already b y Rupp ert [5]. The second one is mu c h more subtle and w e will discuss this p oin t in the general case in the last section, see in particular Prop osition 4.6 . No w we return to the general case n ≥ 2. Let V ( P ) b e the affine hypersurface defined by P = 0 in the a ffine space K n . Let E b e an affine plane in K n suc h that V ( P ) ∩ E is a curv e in E . O ne ma y ask ”Is ther e some r elation b etwe en the numb e r of irr e ducible c om p onents of V ( P ) an d V ( P ) ∩ E ? or, more precisely: A r e these numb ers a l w ays e qual?” The answe r is to suc h questions dep ends on the c hoice of E . Let us lo ok at t w o examples. Example 2.5. (i) Consider the Whitney umbrella S : x 2 − z y 2 = 0, an irreducible singular surface in C 3 . Choose tw o planes E 0 : z = 1 and E 1 : y = 1. One can see that S ∩ E 0 is the union of t wo lines, namely x 2 − y 2 = 0, and S ∩ E 1 is irreducible and isomorphic to C . (ii) Consider the smoot h ir r educible surface S ′ : x 2 y − x − z = 0 in C 3 . Cho o se t w o planes E 0 : z = 0 and E 1 : z = 1. One can see that S ′ ∩ E 0 has t w o comp o nen ts x = 0 and xy − 1 = 0, while S ′ ∩ E 1 is irreducible, and isomorphic to C ∗ . By Bertini’s second Theorem w e kno w that the num b er of irreducible comp o nents of V ( P ) and of V ( P ) ∩ E coincide if the the 2 - plane E is generic, see [1], subsection 9.1.3 for an ex cellen t surv ey of this problem as w ell as Section 5 in [3], for relations to an effec tiv e Hilb ert irreducibilit y theorem. In practice it is quite difficult to decide when a giv en plane E is generic. In the next section we e xplain the relation b et we en this genericit y and transv ersalit y to some Whitney r egula r stratifications, but this is not easy t o c hec k on explic it examples . Moreo ver, once we hav e the factorization of P in the plane E (i.e. in a p olynomial ring in t wo v a r ia bles), it is a second difficult task to reco ver the factorization of P in the p o lynomial ring C [ X 1 , ..., X n ]. This sho ws the need of ha ving an ex tension of Gao’s Theorem for n > 2 v aria bles, and this will be our main result b elo w. 3. Hypersurf ace complements In this sec tion P ∈ C [ X 1 , ..., X n ] is a reduced p o lynomial and P = Q s i =1 P i is the factorization of P into irreducible factors in C [ X 1 , ..., X n ]. The n the asso ciated affine h yp ersurface V ( P ) ⊂ C n has s irreducible comp o nen ts. 4 HANI SH AKER First w e recall a basic result, relating the n um b er s of irreducible factors to the top ology of the h yp ersurface complemen t M ( P ) = C n \ V ( P ) . Prop osition 3.1. s = dim H 1 ( M ( P ) , C ) . Pr o of. Using Corollary 1.4 on p.103 in [2], w e get H 1 ( M ( P ) , Z ) = Z s . The n we use the usual iden tification H 1 ( M ( P ) , C ) = Hom( H 1 ( M ( P ) , Z ) , C ).  Using this r esult, w e can giv e the follow ing description of the generic 2-planes E . Let V ( P ) ⊂ P n b e the pro jectiv e closu re of the hy p ersurface V ( P ) . Then E is said to b e g e ometric al ly gene ric with r esp e c t to V ( P ) if its pro jectiv e closure E is transv ersal to e v ery strata of a Whitney stratification of V ( P ). Applying t he Zariski Theorem of Lefsc hetz t yp e, se e for instance [2], p. 25, w e get the following. Corollary 3.2. L et E b e a ge o m etric al ly generic affin e 2-plane with r esp e ct to the affine hyp ersurfac e V ( P ) . Then V ( P ) and V ( P ) ∩ E have the sam e numb er o f irr e ducible c omp onents. Pr o of. The Z ariski T heorem of Lefsc hetz type implies t ha t the t w o complemen ts M ( P ) and E \ ( V ( P ) ∩ E ) hav e is omorphic fund amen tal g roups. Since w e kno w that, for any path connected space X , the ab elianization ab ( π 1 ( X )) of the fundamen ta l group coincides to the in tegral first homology group H 1 ( X , Z ), the result follo ws using Prop osition 3.1.  F or an y n -tuple A = ( A 1 , ..., A n ) ∈ C [ X 1 , ..., X n ] n of p olynomials, consider the rational 1-fo rm ω ( A ) = n X i =1 A i P dX i defined on the a ffine op en set M ( P ). Such a form ω ( A ) is closed by definition if dω ( A ) = n X i =1 " n X j =1 , j 6 = i  A i P  X j dX j # ∧ dX i = 0 where the subscript X j means taking the partial deriv ativ e with resp ect to X j . In other w ords, the f o llo wing equations should be satisfied. (3.1)  A j P  X i −  A i P  X j = 0 for all i, j = 1 , .., n with i < j . Consider the v ector space F ( P ) of all solutions A = ( A 1 , ..., A n ) ∈ C [ X 1 , ..., X n ] n of the equations (3.1) with the follo wing m ulti- degree b ounds m ultideg ( A i ) ≤ ( m 1 , ..., m i − 1 , ..., m n ) TOPOLOGY AN D F A CTORIZA TION OF POL YNO MIALS 5 for all i = 1 , ...n. Here m ultideg ( P ) = ( m 1 , ..., m i , ..., m n ), and this ob viously means that deg X i P = m i for all i = 1 , ...n exactly as in Definition 2.1. An y closed form ω ( A ) giv es rise to a cohomology class [ ω ( A )] ∈ H 1 ( M ( P ) , C ), if w e w ork with the de Rham cohomology groups of the affine smo oth v ariet y M ( P ) . Theorem 3.3. The line ar map T : F ( P ) − → H 1 ( M ( P ) , C ) define d by T ( A ) = [ ω ( A )] is an isomorphism. In p articular dim F ( P ) = s. Pr o of. T o pro ve the surjec tivit y of the map T , w e recall that a basis for the first de Rham cohomology gr o up H 1 ( M ( P ) , C ) is giv en b y the rational 1-forms (3.2) dP j P j = ω ( B j ) for j = 1 , ..., s , where B j = ( B j 1 , ..., B j n ) with B j i = P · ( P j ) X i P j where the subscript X i indicates the partia l deriv ativ e with res p ect to X i . It is clear that B j ∈ F ( P ), whic h yields the surjectivit y of T . T o pro v e the injectivit y of T , assume that T ( A ) = 0,i.e. (3.3) ω ( A ) = dα for some ratio nal function α ∈ Ω 0 ( M ) . W e can restrict to the case when α is a rational function in view of Grothendiec k Theorem [4 ] sa ying that fo r an affine smo oth v ariety the cohomolog y can be computed using the regular (algebraic) de Rham complex. It follo ws that α is then a regular function of the form α = Q P k , where k ≥ 0 and Q is not divisible b y P . Then fo r any index j ∈ { 1 , 2 , ..., s } , α has a p ole of order k j ≥ 0 along the irreduc ible comp o nent V ( P j ) . W orking lo cally in the neigh b orho o d of a smo o t h p oint p j of V ( P j ), w e see that dα has either a p o le of o rder zero along V ( P j ) if k j = 0 , or a p ole of order k j + 1 if k j ≥ 1 . Hence in an y case w e do not get a p ole of order 1. On the other hand, b y definition, the 1-fo rm ω ( A ) has p oles of order at most one along an y comp onen t V ( P j ). The equalit y (3.3) is p ossible o nly if these po le orders are all zero. This o ccurs only if the p olynomial P divid es all the p olynomials A j for j = 1 , ..., n . But this is imp ossible in view of the m ulti- degree b ounds imp osed on A j , unless all A j are zero.  4. Finding the irreducible f a ctors of P In this section w e e xplain how t o find the irreducible factors of P . Our approa ch is similar to t ha t of Gao explained in [1], ( 9.2.10)-(9.2 .1 2), but w e pay more atten tion to a degenerate case that ma y o ccur, whic h explains our next definition. 6 HANI SH AKER Definition 4.1. W e sa y that a polynomial P ∈ C [ X 1 , ..., X n ] is X 1 -generic if the restriction of the pro jection π 1 : C n → C n − 1 , ( x 1 , x 2 , ..., x n ) 7→ ( x 2 , ..., x n ) to the h yp ersurface V ( P ) has finite fib ers. This prop erty , whic h replaces the condition gcd ( P , P X 1 ) = 1 in Ga o ’s approac h in [3], can be tested b y computer since we ha v e the follo wing ob vious result. Lemma 4.2. L et P = a 0 X m 1 + a 1 X m − 1 1 + ... + a m wher e the c o efficients a j ar e p olynomials in C [ X 2 , ..., X n ] . Then P is X 1 -generic if and only if the ide al sp an ne d by a 0 , ..., a m c oincides to the who l e ring C [ X 2 , ..., X n ] . Example 4.3. (i) If d is the total degree of P and if the monomial X d 1 o ccurs in P with a non- zero co efficien t, then clearly the polynomial P is X 1 -generic. Starting with a n y p olynomial P , w e can arrive at this situation b y making a linear co ordinate c hang e ˜ X 1 = X 1 , ˜ X j = X j + c j · X 1 for j > 1 and suitable constan ts c j ∈ C . (ii) Let n = 3 and consider the p olynomial P = X 2 Y 2 Z 2 + X . Then P is X generic, but not Y -g eneric. W e assume in the sequel that t he p olynomial P is X 1 -generic and define the follo wing tw o asso ciated v ector spaces. Let E ( P ) = pr 1 ( F ( P )), where pr 1 : C [ X 1 , ..., X n ] n → C [ X 1 , ..., X n ] denotes the pro jection on the first factor. Let E ( P ) b e the image of E ( P ) under the canonical pro jection p : C [ X 1 , ..., X n ] → Q ( P ), where w e in tro duce the quotien t ring Q ( P ) = C [ X 1 , ..., X n ] / ( P ). Prop osition 4.4. If the p olynomial P is X 1 -generic, then the fol low ing hold. (i) gc d ( P , P X 1 ) = 1 , wher e t he subscript X 1 indic ates the p artial derivative with r esp e ct to X 1 . (ii) dim E ( P ) = s. Pr o of. T o pro v e (i), it is enough to show that any irreducible fa cto r P k of P does not divide P X 1 . Now, with the not a tion from the pro o f of Theorem 3.3 , we ha v e (4.1) P X 1 = X j =1 , s B j 1 . In this sum, all the terms are divisible b y P k , except p o ssibly B k 1 = P · ( P k ) X 1 P k . This term is divisible b y the irreducible p olynomial P k exactly when ( P k ) X 1 = 0 (oth- erwise deg X 1 P k > deg X 1 ( P k ) X 1 ). But ( P k ) X 1 = 0 implie s that P k ∈ C [ X 2 , ..., X n ] and then, for an y b ∈ C n − 1 suc h t ha t P k ( b ) = 0 (whic h exists since deg P k > 0), the TOPOLOGY AN D F A CTORIZA TION OF POL YNO MIALS 7 line π − 1 1 ( b ) is con tained in the hy p ersurface V ( P ). This contradicts the hy p othesis that P is X 1 -generic, and t hus pro ves (i). T o pro v e (ii), it is enough to show that the classes of the elemen ts B j 1 for j = 1 , ..., s are linearly indep enden t in Q ( P ). Ass ume there is a relation X j =1 , s c j · B j 1 = C · P where c j ∈ C and C ∈ C [ X 1 , ..., X n ]. Chec king as abov e the divisibilit y b y P k , it follo ws that t he co efficien t c k has to v anish, for all k = 1 , ..., s.  Exactly as in the pro of ab ov e, one can sho w that the classes of the eleme n ts (4.2) C j 1 = B j 1 · P X 1 for j = 1 , 2 , ..., s are linearly independent in Q ( P ). It fo llows that the linear subspace (4.3) ˜ E ( P ) = { [ v · P X 1 ] | v ∈ E ( P ) } in Q ( P ) is s -dimensional. Let S : ˜ E ( P ) → E ( P ) b e the inv erse of the linear isomorphism E ( P ) → ˜ E ( P ) sending [ v ] to [ v · P X 1 ] for j = 1 , 2 , ..., s . Note that in the quotien t ring Q ( P ) one has (4.4) [ B i 1 ] · [ B j 1 ] = 0 for i 6 = j and (4.5) [ B i 1 ] · [ B i 1 ] = [ P X 1 ] · [ B i 1 ] for all i = 1 , ..., s. L et v ∈ E ( P ) and write [ v ] = P j =1 , s λ j [ B j 1 ] in Q ( P ). Consider the linear mapping φ v : Q ( P ) → Q ( P ) induced by the m ultiplication by v . Then the equations (4.2 ), (4.4), (4.5) imply that φ v ( E ( P )) ⊂ ˜ E ( P ). It follo ws that ψ v = S ◦ φ v as a linear endomorphism of the s -dimensional vec tor space E ( P ). W e also get ψ v ([ B i 1 ]) = λ i [ B i 1 ] for all i = 1 , ..., s. Remark 4.5. A k ey point here is that the v ector sp ace E ( P ) and the endomorphism ψ v : E ( P ) → E ( P ) can b e computed without kno wing the factorization of P . W e ha v e the following basic result. 8 HANI SH AKER Prop osition 4.6. If the p olynomial P is X 1 -generic and al l the eigenvalues of the endomorphism ψ v : E ( P ) → E ( P ) ar e distinct, say λ 1 , ..., λ s , then, up-to a r e- indexing of the fa c tors, on e ha s P i = g cd ( P , v − λ i P X 1 ) for i = 1 , ..., s. Pr o of. Using the a b o v e equations, w e get a p olynomial C 1 ∈ C [ X 1 , ..., X n ] suc h that (4.6) v − λ i P X 1 = X j 6 = i ( λ j − λ i )[ B j 1 ] + C 1 · P . It follows that the irreducible p olynomial P i divides v − λ i P X 1 . Moreo v er, exactly as in the pro of of Pro p osition 4.4, we see that the irreducible p olynomial P k do es not divide v − λ i P X 1 for k 6 = i .  Reference s [1] G. Ch ` e ze, A. Galligo : F our lec tur es on po lynomial abso lute factorization, Solving Polynomial Equations (Algorithms and Computation in Mathematics, V olume 14). Springer, 2005. [2] A. Dimca : Singularities and T o po lo gy of Hyp ersur faces. Springer, 1992. [3] S. Gao: F actoring multiv ariate p olyno mia ls via partial differential equations, Math. Comp. 72(200 3), 801 -822. [4] A. Grothendiec k: On the de Rham cohomolog y of alg ebraic v arieties, Publ, Math. IHES 29 (1966) [5] W. Ruppert: Reduzibilit¨ a t ebener K urven, J. reine angew. Math. 369(1986 ), 167- 191. [6] W. Rupp ert: Reducibilit y of p olyno mia ls f ( x, y ) mo dulo p , J. Number Theory 77(1999 ),62-70. School of Ma thema tical Sciences, GCU, 68-B New Muslim Town, Lahore P ak- ist a n. E-mail addr ess : h ani.ue t@gmail .com