Fast algorithms for polynomials and matrices Part 6: Polynomial factorization

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1 Fast algorithms for polynomials and matrices Part 6: Polynomial factorization by Grégoire Lecerf Computer Science Laboratory & CNRS École polytechnique Palaiseau Cedex France 1 Classical types of decompositions separable: factorswithequalrootmultiplicities irreducible: setofirreduciblefactors squarefree: equalmultiplicitiesoftheirreduciblefactors absolute: irreducibleoverthealgebraicclosure Separable factorization Seidenberg condition P Squarefree factorization Absolute factorization Univariate factorization Irreducible factorization Basic arithmetic for polynomials and matrices 2

2 Separable decomposition Definition 1. Let K be a field of characteristic p, andletf K[y], withd deg F>0. F is said to be separable if it has no multiple root in the algebraic closure K of K, equiv.discriminant F = Resultant (F,F ) 0. The separable decomposition of F is the decomposition of F according to the multiplicities of its roots: F (y)= s i=1 G i (y qi ) mi,with G i (y qi ) pairwise coprime, G i separable, q i is a power of p if p>0, otherwiseq i =1, m i is not divisible by p, (q i,m i ) pairwise distinct. Proof. The roots of G i (y qi ) are the ones of F with multiplicity q i m i. Gianni and Trager (1996) showed that the separable decomposition of a polynomial f K[x] of degree d can be computed in softly quadratic time over any field, by extending Musser s algorithm. A similar extension of Yun s algorithm leads to: Proposition 2. [ Lecerf, 2008]The separable decomposition of a polynomial f K[x] of degree d can be computed with O(M(d) log d). 3 Example of separable decomposition Mmx] use "factorix"; x == polynomial (0,1); F == (x^9 + x^3 + 1)^2 * (x^4-1); Mmx] separable_factorization F [[x 4 +1, 1, 1], [x 9 + x 3 +1, 1, 2], [1, 1, 1]] Mmx] sep == separable_factorization (F mod modulus 3) [[x +2, 1, 7], [x 3 + x 2 + x +1, 1, 1], [x 2 + x +2, 3, 2]] Mmx] tmp == [ dilate (f[0], f[1])^f[2] f in sep ]; F mod modulus 3 = big_mul tmp true 4

3 Reductions to the separable case The squarefree factorization reduces to splitting a separable polynomial F (y p ) into G(y p )H(y) p,withg(y p ) and H(y) squarefree, assuming that F is separable. This task is not computable in general! It is computable if we assume that the Seidenberg condition P holds, that is: p-th roots can be extracted in any (purely inseparable) algebraic extension of K. Example 3. If K = F p then we always have F (y p )=F (y) p,hencethatg =1 and H = F. Example 4. If K = F p k then we always have F (y p )=H(y) p,whereh = d i=0 F 1/p i y i.theextractionof the p-th root of a F p k can be computed as a pk 1,whichtakesO(klog p) operations in F p k. ( Example 5. If K = F p (x) then gcd F (x, y), ) F (x, y) (x, y p )=H(x, y) p. x Let F be a separable irreducible polynomial. Then F (y p ) is either irreducible or a power of p. Theirreducible factorization can be deduced from the separable decomposition by the only use of p-th root extractions. 5 Irreducible factorization Theorem 6. [ van der Waerden (1930), Fröhlich and Shepherdson (1956)] The irreducible decomposition of a univariate polynomial over an effective field is not computable in general. AfieldK is explicitly finitely generated over a field F if it is the fraction field of F[x 1, prime and explicitly given by a finite set of generators.,x n ]/P with P Theorem 7. [ van der Waerden, Fröhlich and Sheperdson, Seidenberg, Richman,...] The irreducible decomposition is computable over any explicitly finitely generated extension of a prime field. General algorithms Davenport and Trager (1981): never fully implemented. Steel (2005): thefirst(andstillunique)mostgeneralalgorithm,implementedinthemagma computer algebra system. 6

4 Algorithmic dependencies for irreducible factorizations Under the assumption that the polynomial to be factored is separable of degree d. Specific algorithms for F pk[y], where F pk is the finite field with pk elements. Factorization in F p[y] implies factorization in Q[y]. Factorization in K[y] implies factorization in K[α][y], with α separable. Factorization in K[y] implies factorization in K[x, y]. Factorization in K[x, y] implies factorization in K[x1,., xn, y]. 7 Short history about univariate factorization over finite fields Early ideas by Gauss, Galois, Arwins. First algorithms: Berlekamp (1970), Zassenhaus (1969), Cantor & Zassenhaus (1981). Best practical algorithm : von zur Gathen & Shoup (1992) in O (d2 + d log q) operations in F q and degree d. Several optimized variants for specific cases: von zur Gathen, Shoup, Niederreiter, Gao, Kaltofen,... The best known theoretical algorithm is a combination of: Kaltofen and Shoup (1998): reduction to modular composition, Kedlaya and Umans (2008): softly linear time for modular composition over finite fields. Randomized bit complexity in O ((d1.5 + d log q)log q). Modular composition: g(h) mod f, with f, g, h in K[x] of degree at most d. Mmx] use "gregorix"; x == polynomial (0,1); F == (x^9 + x^3 + 1)^2 * (x^4-1); Mmx] irreducible_factorization (F mod modulus 3) [[x + 2, 7], [x + 1, 1], [x2 + 1, 1], [x2 + x + 2, 6]] 8

5 Univariate factorization over Q First exponential time algorithm by Kronecker (1882): constructandtestallthepossibilitiesaccording to the divisibility constraints. Hensel lifting popularized in computer algebra by Zassenhaus (1969), with exponential time. Mmx] use "gregorix"; p == modulus 5; F == polynomial (-1, 0, 0, 0, 1) x 4 1 Mmx] irreducible_factorization (F mod p) [[x +1, 1], [x +2, 1], [x +3, 1], [x +4, 1]] Mmx] p_adic_precision:= 4; Fp: Polynomial P_adic Integer == F; Mmx] fp == irreducible_factorization (Fp) [[(1 + O(5 4 )) x O(5 4 ), 1], [(1 + O(5 4 )) x O(5 4 ), 1], [(1 + O(5 4 )) x +1+O(5 4 ), 1], [(1 + O(5 4 )) x O(5 4 ), 1]] Mmx] fp[0][0] * fp[3][0] 9 (1 + O(5 4 )) x 2 + O(5 4 ) x +1+O(5 4 ) First polynomial time algorithm by Lenstra & Lenstra & Lovász (1982) Mmx] F == polynomial (6, 0, -5, 0, 1) x 4 5 x 2 +6 Mmx] N z == z - evaluate (F, z) /evaluate (derive F, z); Mmx] a == N N N N N Mmx] integer_relation ([1, a], 30) [ e8, e8] Mmx] c == integer_relation ([1, a, a^2], 30) [ , 0, ] Mmx] G == polynomial (as_integer c[i] i in 0..#c) x 2 +2 Mmx] F mod G 10

6 0 Mmx] F div G x 2 +3 Remarks: Here we are using PSLQ (partial sums of squares / lower trapezoidal orthogonal decomposition) by Bailey & Ferguson (1991). The approach suffers from the need of computing big integer relations never competitive in practice. The first practical polynomial time algorithm is due to van Hoeij (2002). Implementedandimproved by Belabas & van Hoeij & Klüners & Steel (2004), andfurtherimprovedbynovocin. F = F 1 F r F = F F 1 F F F r F r Mmx] use "gregorix"; p == modulus 9973; F == swinnerton_dyer_polynomial 2 * swinnerton_dyer_polynomial 3; irreducible_factorization (F mod p) [[x x + 342, 1], [x x , 1], [x x +1, 1], [x x +1, 1], [x x + 342, 1], [x x , 1]] 11 Mmx] Fp == F :> Polynomial P_adic Integer; fp == irreducible_factorization (Fp); Mmx] G == [(Fp div fp[i][0]) * derive fp[i][0] i in 0..#fp]; Mmx] W == [ evaluate (G[i], 3 :> P_adic Integer)[3,20] i in 0..#G ] [ , \ , \ , \ , \ , ] Mmx] bit_precision:= 1000; I == integer_relation ([ W[i] :> Floating i in 0..#G] >< [ ^17 ], 200); [ as_integer I[i] i in 0..#I ] [0, 1, 0, 0, 1, 0, 1] Mmx] Hp == fp[1][0] * fp[4][0] (1 + O( )) x 4 + O( ) x 3 +( O( )) x 2 + O( ) x +1+O( ) 12

7 Mmx] to_integer u == if u[0,3] > 9973^3 div 2 then u[0,3] ^3 else u[0,3]; H == polynomial (to_integer Hp[i] i in 0..#Hp) x 4 10 x 2 +1 Mmx] F div H x 8 40 x x x Advantage to the first polynomial time method: smaller integer relations! Very efficient in practice! 13 Irreducible factorization over an algebraic extension If α is separable over K then the factorization in K[α][y] follows from the one in K[y] and primitive element calculations [van der Waerden, Trager]. Otherwise we can assume that α p = a K\K p without loss of generality. This situation can be avoided if we reorganize the presentation of K[α] over its prime field [van der Waerden, Maclane (1930)]. Otherwise one can factor F (α, y) p in K[y p ] and then rely on Seidenberg condition P. 14

8 d x :=partial degree in x d y partial degree in y Bivariate polynomial factorization Theorem 8. Factorization of separable polynomials in K[y] implies Factorization of separable polynomials in K[x][y]. First algorithm goes back to Kronecker: exponentialtime,andreductiontounivariatefactorizationin degree d x d y. Hensel lifting and recombination with exhaustive search. Reduction to univariate factorization in degree d y. Studied by Musser (1973), Wang & Rothschild (1975), Wang (1978), von zur Gathen (1984), Bernardin (1999), Gao & Lauder (2000). Mmx] use "gregorix"; p == modulus 101; x == polynomial (0,1) mod p; F == polynomial (-x^2 - x^4, 2 + 2*x^2, -1, -2, 1) y y y 2 +(2x 2 +2)y x x 2 Mmx] Fs == F :> Polynomial Series Modular Integer; fs == irreducible_factorization (Fs) 15 [[(1 + O(z 10 )) y z z z z 8 + O(z 10 ), 1], [(1 + O(z 10 )) y z z z z 8 + O(z 10 ), 1], [(1 + O(z 10 )) y + 50 z z z z 8 + O(z 10 ), 1], [(1 + O(z 10 )) y z z z z 8 + O(z 10 ), 1]] Mmx] Gs == fs[0][0] * fs[1][0] (1 + O(z 10 )) y 2 + O(z 10 ) y z 2 + O(z 10 ) Mmx] G == polynomial (Gs[i][0,4] i in 0..#Gs) y x Mmx] F mod G 0 Mmx] F div G y y + x 2 First polynomial time algorithm due to Kaltofen (1982). Several variants by Lenstra, Kannan, Lovász, Chistov, Grigoriev, von zur Gathen,... all derived from LLL and Padé-Hermite. Mmx] series_precision := 10; s == -fs[0][0][0] 16

9 z z z z 8 + O(z 10 ) Mmx] M == pade_hermite ([s^0, s^1, s^2], 10) 19 x x x x x x x x 2 41 x x 2 86 x x 2 Mmx] polynomial (@(row (M, 0) / (82 mod p))) y x First softly quadratic time algorithm by Gao (2003). Fast recombination stage for Hensel lifting by Belabas, van Hoeij, Klüners, and Steel (2004), with worst case precision Ω(d x d y ),andtotalcostinõ(d x d y 3 ). F = F 1 F r F = F F 1 F F F r F r Mmx] Gs == [(Fs div f[0]) * derive f[0] f in fs] 17 [(1 + O(z 10 )) y 3 +( z z z z 8 + O(z 10 )) y 2 +(2+2z z z z 8 + O(z 10 )) y z z z z 8 + O(z 10 ), (1 + O(z 10 )) y 3 +( z z z z 8 + O(z 10 )) y 2 +( z z z 8 + O(z 10 )) y + z z z z 8 + O(z 10 ), (1 + O(z 10 )) y 3 +( z z z z 8 + O(z 10 )) y 2 +( z 2 + O(z 10 )) y z z z 6 +7z 8 + O(z 10 ), (1 + O(z 10 )) y 3 +(50 z z z z 8 + O(z 10 )) y 2 + ( z 2 + O(z 10 )) y + 51 z z z z 8 + O(z 10 )] Mmx] M == [ g[i][4] g in Gs i in 0..4 ] Mmx] ker M Mmx] series_precision:=10; Gs[0] + Gs[1] (2 + O(z 10 )) y 3 +(97 + O(z 10 )) y 2 +(2z 2 + O(z 10 )) y + O(z 10 ) 18

10 Mmx] fs[0][0] * fs[1][0] (1 + O(z 10 )) y 2 + O(z 10 ) y z 2 + O(z 10 ) Recombination scheme with optimal precision d x +1: Lecerf (2007), with cost in Õ(d x d y 2 ) forany characteristic. Let F = F 1 F s be the irreducible factorization in K[[x]][y] of F. Let G i = F i F /F i dx+1,fori in {1,,s}. Let ζ be the residue class of y modulo F. The recombinations can be read off from the echelon solution basis of: (l 1,,l s ) K s s d l i G i (x, ζ) =0, where F (x, ζ)=0. dx i=1 F y Mmx] H == [polynomial (g[i][0,5] i in 0..#g) g in Gs] [y 3 +(38 x x ) y 2 +(25 x 4 +2x 2 +2)y + 50 x x 2,y 3 +(63 x x ) y 2 + (76 x ) y + 51 x 4 + x 2,y 3 +(38 x x ) y 2 +(100 x ) y + 12 x x 2 +2, y 3 +(63 x x 2 ) y 2 +(100 x ) y + 89 x x 2 ] Mmx] Dx h == polynomial (derive h[i] i in 0..#h); 19 Mmx] Dy h == derive h; Mmx] D h == (Dx h * Dy F - Dy h * Dx F) * Dy F - (Dx Dy F * Dy F - Dy Dy F * Dx F) * h; Mmx] R == [ (D h) mod F h in H ] [(97 x x 5 ) y 3 +(24 x x 5 ) y 2 +(83 x x x 5 ) y +6x 11 +6x x x 5, (4 x 7 +6x 5 ) y 3 +(77 x x 5 ) y 2 +(18 x 9 +9x 7 +3x 5 ) y + 95 x x 9 + x 7 +3x 5, (4 x x 5 ) y 3 +(12 x x 5 ) y 2 +(83 x x x 5 ) y +6x x x x 5, (97 x 7 +6x 5 ) y 3 +(89 x x 5 ) y 2 +(18 x x 7 +6x 5 ) y + 95 x x x x 5 ] Mmx] N == matrix (h[i][5] h in R i in 0..4) Mmx] ker N

11 Multivariate polynomial factorization Let F K[x 1,,x n ] Either extend the previous techniques from K[[x]][y] to K[[x 1,,x n1 ]][x n ], or reduce to K[x, y] via the Bertini/Hilbert theorem: for a finite subset S of K upper bound the density of points (a,b,c) in (S n ) 3 for which F (a 1 x+b 1 y +c 1,, a n x + b n y + c n ) is irreducible if F is irreducible. Hilbert (1892) (before Bertini): the density tends to 0 for large S. Heintz & Sieveking (1981), Kaltofen (1982): use in computer algebra. von zur Gathen (1985): 9 d2 /#S. Bajaj, Canny, Garrity & Warren (1993): d 4 /#S when K = C. Kaltofen (1995): 2d 4 /#S when K is perfect. Gao (2003): 2d 3 /#S when p =0 or p 2d 2. Lecerf (2007): 23 8 d2 /#S when p =0 or p d(d 1) + 1. Softly linear reduction to one variable whenever n 3, assumingfastmultivariatepowerseriesproduct available. 21