Journal of Mathematical Sciences, Vol. 134, No. 5, 2006

COMPLEXITY BOUND FOR THE ABSOLUTE FACTORIZATION OF PARAMETRIC POLYNOMIALS A. Ayad∗

UDC 510.52, 512.622

An algorithm is constructed for the absolute factorization of polynomials with algebraically independent parametric coefficients. It divides the parameter space into pairwise disjoint pieces such that the absolute factorization of polynomials with coefficients in each piece is given uniformly. Namely, for each piece there exist a positive integer l ≤ d, l variables C1 , . . . , Cl algebraically independent over the ground field F , and rational functions bJ,j of the parameters and of the variables C1
, . . . , Cl such that for any parametric polynomial f with coefficients in this piece, there exist c1 , . . . , cl ∈ F with f = j Gj , where Gj = |J| BJ,j Z J is absolutely irreducible. Here Z = (Z0 , . . . , Zn ) are the variables of f , each BJ,j is the value of bJ,j at the coefficients of f and c1 , . . . , cl , and F denotes the algebraic 2 closure of F . The number of pieces does not exceed (2d2 +1)2n+3d+5 , and the algorithm performs dO(ndr ) arithmetic n+1+d  operations in F (thus the number of operations is exponential in the number r = n+1 of coefficients of f ), and its binary complexity is bounded by dO(ndr

2)

if F = Q and by



pdndr

2

O(1)

if F = Fp , where d is an upper bound

on the degrees of polynomials. The techniques used include the Hensel lemma and the quantifier elimination in the theory of algebraically closed fields. Bibliography: 20 titles.

1. Introduction and notation The problem of factorization of multivariate polynomials with coefficients in a field was studied by many mathematicians. Kronecker worked on this problem in the case of polynomials with integer coefficients. An exposition of Kronecker’s algorithm and a proof that its complexity is exponential can be found in [12] and [15]. In 1967, Berlekamp described an algorithm that factorizes a univariate polynomial with coefficients in a finite field of q elements and has the polynomial complexity O(d3 q), where d is the degree of this polynomial (see, e.g., [2, 3, 15]). Using the algorithm by Berlekamp and the Hensel lemma (see, e.g., [5, 7, 10, 15, 17, 18, 20]), Zassenhaus (see, e.g., [15, 20]) designed an algorithm for factorization of polynomials in Z[X], still having an exponential complexity in the size of the input. The first algorithm for univariate factorization over Q with a polynomial complexity in the size of the input polynomial was published in [14] by Lenstra, Lenstra, and Lovasz. In 1982, Grigoriev [7] and Chistov [5] described a polynomial-time algorithm for factorization of polynomials in several variables with coefficients in a field that is a finite extension of a purely transcendental extension of its prime subfield. A multivariate polynomial with coefficients in an extension of F is called absolutely irreducible if it is irreducible over the algebraic closure F of F . The absolute factorization of a multivariate polynomial is its decomposition into absolutely irreducible factors. For more details on algorithms for absolute factorization of polynomials, one can consult the paper [5] by Chistov and the thesis [19] by Ragot. In [5], Chistov presented a polynomial-time algorithm for the reduction of absolute factorization to factorization over the ground field. A parametric polynomial is a polynomial whose coefficients are polynomials (over F ) in the parameters. In this paper, we restrict our attention to the case where all coefficients of parametric polynomials are algebraically independent; see Remark 1.1 for a discussion of the general case. Let d ∈ N \ {0} and Fd [Z0 , . . . , Zn ] = {f ∈ F [Z0 , . . . , Zn ], deg(f) = d} bethe set of all polynomials degree  of n+d n+d exactly d. Then there exists a bijection between this set and the set P = F ( n ) \ {(0, . . . , 0)} × F (n+1) of coefficients of these polynomials, which will be called the parameter space; here the first factor corresponds to monomials of degree  d, and the second one corresponds to monomials of degree less than d. This bijection sends a polynomial f = |I|≤d aI Z I ∈ Fd [Z0 , . . . , Zn ] to the tuple (aI )|I|≤d ∈ P, where I = (i0 , . . . , in ) ∈ Nn+1 , |I| = i0 + · · · + in is the weight of the (n + 1)-tuples I and Z I = Z0i0 · · · Znin , and the family {aI }|I|≤d is the set of parameters of the parametric polynomial f. The input of the algorithm is the vector of parametric coefficients taking values in the parameter space. Its output consists of the following three parts: ∗ IRMAR,

Campus de Beaulieu, Universit´e de Rennes 1, France, e-mail: [email protected].

Published in Zapiski Nauchnykh Seminarov POMI, Vol. 316, 2004, pp. 5–29. Original article submitted December 2, 2004. c 2006 Springer Science+Business Media, Inc. 1072-3374/06/1345-2325 

2325

(1) A decomposition of the parameter space into pairwise disjoint pieces of the form  P= Vi , 1≤i≤(2d2+1)2n+3d+5



where stands for the disjoint union. (2) Each piece Vi is represented as a constructible set defined by a quantifier-free formula of the first-order (β) (β) theory of algebraically closed fields, i.e., the algorithm produces a finite family of polynomials Bi,δ , Ci ∈

 (β) (β) F [{aI }|I|≤d ] such that Vi = β = 0) . δ (Bi,δ = 0) ∧ (Ci (3) For each piece Vi , the absolute factorization of polynomials with coefficients in Vi is given uniformly, i.e., there exist a positive  integer li ≤ d, li variables C1 , . . . , Cli algebraically independent over F , and rational (i) functions bJ,j ∈ F {aI }|I|≤d , C1 , . . . , Cli such that for any polynomial f with coefficients (aI )|I|≤d ∈ Vi , there  
 (i) exist c1 , . . . , cli ∈ F with f = j Gj and Gj = |J| bJ,j {aI }|I|≤d , c1 , . . . , cli Z J absolutely irreducible. Note that there exists a bounded number of possibilities for c1 , . . . , cli (see Remark 7.4). The main result of this paper is an explicit description of all pieces Vi (Corollary 6.2) and rational functions (i) bJ,j from part (3) of the output of the algorithm (Theorem 7.3), and a complexity bound for this algorithm (Sec. 8). Remark 1.1. In the case where the parameters are not independent, e.g., they are contained in a constructible set V ⊂ P given as above, we consider the partition    V = Vi ∩ V . 1≤i≤(2d2+1)2n+3d+5

Each set Vi ∩ V can be found by the algorithm of [5, 7], and the absolute factorization of parametric polynomials from Vi ∩ V can be represented as in (3). In this paper, we study the complexity of this algorithm only in the case where the parameters are independent (Sec. 8), i.e., we do not consider the complexity of the representation of V . Example. For F = Q, we consider the parametric polynomial f = (a2 + b)X 2 + aXY + bX + aY + b with two variables X and Y , a and b being parameters. We obtain the following decomposition of the parameter space: Q2 = W1  W2  W3 , where W1 = {(a, b) ∈ Q2 , a2 + b = 0}, and for each (a, b) ∈ W1 , we have the absolute factorization f = (X + 1)(aY + b); W2 = {(a, b) ∈ Q , a + b = 0, a = 0}, and for each (a, b) ∈ W2 , f is absolutely irreducible; W3 = {(a, b) ∈ Q2 , a = 0, b = 0}, and for each (a, b) ∈ W3 , 2

2

f = b(X − j)(X − j 2 ), where j is a primitive cubic root of unity. The tools used in this paper are the Hensel lemma and the quantifier elimination in the theory of algebraically closed fields [4]. In Secs. 2 and 3, we present an initial decomposition of the parameter space (Corollary 2.2) and describe its complexity, as a preparation for applying the parametric Hensel lemma in Secs. 4 and 5. This lemma and the quantifier elimination will be used in Secs. 5, 6, and 7 to complete the decomposition of the parameter space (Theorem 7.3). The analysis of the total complexity of this algorithm will be carried out in Sec. 8. It would be interesting to obtain a lower bound on the complexity of absolute factorization of parametric polynomials. To the author’s best knowledge, algorithms for factorization of parametric polynomials (and, moreover, studies on the complexity of this problem) have never been published. Let us also mention that an algorithm for solving systems of parametric univariate polynomials was designed in [8]. In a forthcoming paper, we are going to apply the described algorithm to solving parametric systems of multivariate algebraic equations. 2326

2. Preparation for the Hensel lemma Since the Hensel lemma, which will be used below (Lemma 4.1), is applicable only to polynomials f ∈ F [X, Y1 , . . . , Yn ] that are monic as polynomials in X with f0 (X) = f(X, 0, . . . , 0) square-free in F [X], we should partition the parameter space P into pairwise disjoint subsets W such that with each W we can uniformly associate a polynomial g ∈ F ({aI }|I|≤d )[X, Y1 , . . . , Yn ] (i.e., a polynomial whose coefficients are rational functions in the parameters {aI }|I|≤d) such that for each evaluation {aI }|I|≤d ∈ W , lcX (g) = 1 and

g(X, 0, . . . , 0) is square-free in F [X],

where lcX (g) denotes the leading coefficient of g with respect to X as a univariate polynomial in F [Y1 , . . . , Yn ][X]. With an (n + 1) × (n + 1) matrix T with entries in F and a polynomial f ∈ Fd [Z0 , . . . , Zn ], we associate a polynomial gf,T ∈ F [X, Y1, . . . , Yn ] defined by gf,T (X, Y1 , . . . , Yn ) = f(T (X, Y1 , . . . , Yn )). For technical reasons, we suppose, without loss of generality, that |F | ≥ max{2d2 , r} (in the case char(F ) = p > 0). Proposition 2.1. One can explicitly produce a family {T1 , . . . , TN1 } of nonsingular (n + 1) × (n + 1) matrices with coefficients in F such that for any polynomial f ∈ F [Z0 , . . . , Zn ] of total degree d, there exists i, 1 ≤ i ≤ N1 = (d + 1)n , such that the polynomial gf,Ti fulfills the condition 0 = lcX (gf,Ti ) ∈ F .  Proof. Let f = |I|≤d aI Z I ∈ Fd [Z0 , . . . , Zn ] and T = (ti,j )1≤i,j≤n+1 ; then    0 1 n  Xd + h gf,T (X, Y1 , . . . , Yn ) =  aI ti1,1 ti2,1 · · · tin+1,1 i0 +···+in =|I|=d

with degX (h) < d, and 0 = lcX (gf,T ) =



0 1 n aI ti1,1 ti2,1 · · · tin+1,1 ∈ F [t1,1, t2,1 , . . . , tn+1,1]

i0 +···+in =|I|=d

is a homogeneous polynomial in t1,1 , t2,1 , . . . , tn+1,1 of degree d, degti,1 (lcX (gf,T )) ≤ d, 1 ≤ i ≤ n + 1. We fix pairwise distinct elements b0 , . . . , bd ∈ F (if char(F ) = 0, then we take bi = i; if char(F ) = p > 0, then bi ∈ Fpm , where pm−1 ≤ d < pm ). For every fixed f, there exists (t1,1 , t2,1 , . . . , tn+1,1) ∈ {b0 , . . . , bd}(n+1) with lcX (gf,T )(t1,1 , t2,1, . . . , tn+1,1) = 0, as can easily be seen by induction on n. Since lcX (gf,T ) is homogeneous, we can take t1,1 = 1. Since det(T ) = 0, we can take  1 0  t2,1 T =  .. . tn+1,1

 , 

1 ..

.

0

1

where the entries of the matrix T vanish except for the first column and the diagonal. Taking N1 = (d + 1)n completes the proof.
Proposition 2.1 implies that P = 1≤i≤N1 Wi , where Wi = {f ≡ (aI )|I|≤d ∈ P, 0 = lcX (gf,Ti ) ∈ F }. i = Wi \ Wj , 1 ≤ i ≤ N1 ; then Let W 1≤j 0. For any f ≡ (aI )|I|≤d ∈ W nents of X in the monomials in X, Y1 , . . . , Yn with nonzero coefficients of the polynomial lcX (g1f,T ) gf,Ti (X, Y1 , . . . , i Yn ); then t 1 (t) gf,Ti (X, Y1 , . . . , Yn ) = gf,Ti (X p , Y1 , . . . , Yn ) lcX (gf,Ti ) 2327

(t)

with gf,Ti satisfying ∂ (t) (t) (t) (g ) = 0, degX (gf,Ti ) = dp−t , degYj (gf,Ti ) ≤ d, 1 ≤ j ≤ n, ∂X f,Ti (t)

(t)

where gf,Ti is monic as a polynomial in X and the coefficients of gf,Ti are rational functions in the parameters {aI }|I|≤d ; and the factorization 

(t)

gf,Ti =

gk , where gk are absolutely irreducible,

k

provides the absolute factorization of f (using the inverse transformation Ti−1 ). Setting   ∂ (t)   Wi,t = f ∈ Wi , ) = 0 , (g ∂X f,Ti we obtain



i = W

(1)

i,t , W

log

d

0≤t≤ log2 p  2

where ∗ is the integer part of ∗. i = W i,0 , where W i,0 is defined by (1) under the convention that 00 = 1. For Case char(F ) = 0. We obtain W  any f ≡ (aI )|I|≤d ∈ Wi,t , let  (t) DiscX (gf,Ti )

Then

= resX

(t) gf,Ti ,

 ∂ (t) (g ) ∈ F [Y1 , . . . , Yn ]. ∂X f,Ti

degYj (DiscX (gf,Ti )) ≤ d(2dp−t − 1) = d1 , (t)

1 ≤ j ≤ n.

(1) i,t , DiscX (g(t) ) ≡ 0}, W (2) = {f ∈ W i,t , DiscX (g(t) ) = 0}; then Let Wi,t = {f ∈ W i,t f,Ti f,Ti

i,t = W (1)  W (2) . W i,t i,t We fix pairwise distinct elements b0 , . . . , bd1 ∈ F (if char(F ) = p > 0, we take bi ∈ Fpm , where pm−1 ≤ d1 < pm ). (2) (j) (j) (t) For every f ∈ Wi,t , there exists c(j) = (c1 , . . . , cn ) ∈ {b0 , . . . , bd1 }n such that DiscX (gf,Ti )(c(j) ) = 0, as can (t)

(j)

(j)

easily be seen by induction on n; then gf,Ti (X, c1 , . . . , cn ) is square-free in F [X]. (t,j)

(t)

(j)

(j)

(t,j)

For gf,Ti (X, Y1 , . . . , Yn ) = gf,Ti (X, Y1 + c1 , . . . , Yn + cn ), the following holds: gf,Ti is monic as a polynomial (t,j)

(t,j)

(t,j)

in X; gf,Ti (X, 0, . . . , 0) is square-free in F [X]; degX (gf,Ti ) = dp−t ; degY1 ,...,Yn (gf,Ti ) ≤ d, and its coefficients are
(t,j) (t,j) rational functions in the parameters {aI }|I|≤d. Given the decomposition gf,Ti = k hk of gf,Ti into absolutely irreducible factors hk , the formula (t)

(t,j)

(j)

gf,Ti (X, Y1 , . . . , Yn ) = gf,Ti (X, Y1 − c1 , . . . , Yn − c(j) n ) =



(j)

hk (X, Y1 − c1 , . . . , Yn − c(j) n )

k (t)

yields the absolute factorization of gf,Ti . Let (2,j)

Wi,t Then

(2)

(t)

(2)

(2)

Wi,t =

 1≤j≤N2

2328

(t,j)

= {f ∈ Wi,t , DiscX (gf,Ti )(c(j)) = 0} = {f ∈ Wi,t , gf,Ti (X, 0, . . . , 0) is square-free in F [X]}. (2,j)

Wi,t , where N2 = (d1 + 1)n .

(2) (2,j1)  (2,j) = W (2,j) \ To obtain a partition of Wi,t , we set W , 1 ≤ j ≤ N2 . Then i,t i,t 1≤j1 0, bi ∈ Fpm with pm−1 ≤ d1 < pm , then l(bi ) ≤ log2 (pm ) ≤ log2 (pd1 ) ≤ log2 (2pd2 ). The evaluation of the homogeneous the point (t1,1 , . . . , tn+1,1) ∈  lcX (gf,Ti ) of degree  d at  polynomial   n+d {b0 , . . . , bd }n+1 can be executed using n+1+d − 3 multiplications and − 1 additions in F . Hence the n+1 n       i costs C1 = N1 n+1+d + n+d construction of all the W ≤ (d + 2)2n+1 arithmetic operations in F , and its n+1

binary complexity • if F = Q, C1

C1

n

can be bounded as follows:

       n+1+d n+d 2 2 ≤ N1 O(d (log2 d) ) + O(d log2 d) ≤ (d + 2)2n+1 O d2 (log2 d)2 ; n+1 n 2329

• if F = Fp ,

  C1 ≤ (d + 2)2n+1 O (log2 p)2 .

  (t) The complexity of the computation of DiscX (gf,Ti ) is O (2dp−t −1)2.376 ≤ O(d2.376) (see [9]).    (2,j) costs C2 = N2 n+d1 ≤ (2d2 + 1)2n arithmetic operations in F . Its binary The construction of all the W i,t n complexity C2 can be bounded as follows: • if F = Q, C2

        n + d1 2 2 ≤ N2 ≤ (2d2 + 1)2n O d2 (log2 d)2 ; O d (log2 d) + O d log2 d n

• if F = Fp ,

  C2 ≤ (2d2 + 1)2n O (log2 p)2 .

  (t) The complexity of the computation of Gf,Ti is O d2 (dp−t )2 ≤ O(d4 ). 2.376 ) + O(d4 ) ≤ Thus the total complexity of this partition of the parameter space is C ≤ C1 + C2 + O(d O (2d2 + 1)2n+1 . Its binary complexity C  can be bounded as follows: • if F = Q,  

C  ≤ (2d2 + 1)2n+1 O d2 (log2 d)2 ;

• if F = Fp ,

  C  ≤ (2d2 + 1)2n+1 O (log2 p)2 . 4. Hensel lemma

At the second stage, we want to parametrize the Hensel lemma (see, e.g., [5, 7, 10, 15, 17, 18] and [20]). Lemma 4.1. Let RN = F [Y1, . . . , Yn ]/((Y1 , . . . , Yn )N ), where 1 ≤ N ≤ ∞ and R∞ = F [Y1, . . . , Yn ]. Let N > 1, and let g ∈ RN [X] satisfy the following properties: (H1) lcX (g) = 1; (H2) g0 (X) = g(X, 0, . . . , 0) is square-free in F [X]. (1) (s) (1) (s) Then for each decomposition of g0 of the form g0 = g0 · · · g0 , where g0 , . . . , g0 ∈ F [X] are monic, the following assertion holds. For each multi-index I ∈ Nn such that N > |I| ≥ 1, there exist unique polynomials (1) (s) gI , . . . , gI ∈ F [X] that satisfy the following conditions: (j) (j) (i) deg(gI ) < deg(g0 ), |I| ≥ 1, 1 ≤ j ≤ s; (ii) in the completion of RN [X] with respect to (Y1 , . . . , Yn ), one has the decomposition g = G(1) · · · G(s),

where

(j)

G(j) = g0 +



(j)

gI Y I ,

1 ≤ j ≤ s.

|I|≥1

For N  < N , the decomposition (ii) of the image of g under the natural homomorphism ν : RN [X] → RN
[X] is obtained by applying ν. Proof. The last part of the lemma is obvious. To prove the rest, note that for any ideal I ⊆ F [Y1 , . . . , Yn ] = F [Y ], the completion of F [Y ] with respect to (Y1 , . . . , Yn ) equals F [[Y ]]/IF [[Y ]] (see, e.g., [6, p. 179]). This observation with I = (Y1 , . . . , Yn )N and, e.g., [6, Theorem 7.18 and Ex. 7.22]) complete the proof. (Cf. also [5, 7, 17, 18].)
 Let us write g in the form g = g0 + |I|≥1 gI Y I , where gI ∈ F [X], deg(gI ) < deg(g0 ) (this can be done, because lcX (g) = 1). Condition (ii) of Lemma 4.1 is equivalent to gI =





∪0≤l≤s Jl =I 0≤k≤s

2330

(k)

gJk .

Then



gI =

(1)

(j−1) (j) (j+1) gI g0

g0 · · · g0

(s)

· · · g0 + VI ,

|I| ≥ 1,

(2)

1≤j≤s (j)

where VI depends only on the polynomials gJ , |J| < |I|, 1 ≤ j ≤ s. (1) (s) Thus for |I| fixed, the coefficients of gI , . . . , gI form a vector of F d which is the unique solution of the linear system Bx = b obtained by (2), where B is a d × d matrix whose coefficients depend only on the coefficients of (1) (s) (1) (s) g0 , . . . , g0 (B is invertible by the uniqueness of gI , . . . , gI in Lemma 4.1). (1) (2) (1) (2) In the case s = 2, it follows that B = Sylv (g0 , g0 ) is the Sylvester matrix of g0 and g0 .    (j) (j,I) i (j) (j) (j) α i X , g0 = X k j + αi X i , 1 ≤ j ≤ s, where kj = deg(g0 ), deg(g0 ) = d = kj , Set gI = j 0≤i d. We will prove the theorem in the case s = 2, the general case can be considered in a similar way.  (1) (2) If s = 2, then VI = 1≤|J|≤|I|−1 gJ gI−J . The only thing to show is that the equations VI = 0 for d < |I| ≤ 2d (1)

(2)

imply VI = 0 for |I| > d. Lemma 5.1 gives gI = gI = 0, d < |I| ≤ 2d; we will show by induction on t that VI = 0, |I| = d + t, t ≥ d + 1. Indeed, for t = d + 1 (|I| = 2d + 1) we obtain VI =



(1) (2)

gJ gI−J =

1≤|J|≤2d



(1) (2)

gJ gI−J +

1≤|J|≤d



(1) (2)

gJ gI−J ;

d