Fast algorithms for factoring polynomials A selection. Erich Kaltofen North Carolina State University google->kaltofen google->han lu

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1 Fast algorithms for factoring polynomials A selection Erich Kaltofen North Carolina State University google->kaltofen google->han lu

2 Overview of my work 1 Theorem. Factorization in K[x] is undecidable even if K is an effective field [van der Waerden 35, Fröhlich and Shepherdson 55]. Theorem. Factorization in Z p [x] [Berlekamp 67] and Q[x] [LLL] is polynomial-time. If factorization in K[x] is polynomial-time then factorization in K[x 1,...,x n ] is polynomial-time [Kaltofen 82]. If arithmetic in K is polynomial-time then factorization in K[x 1,...,x n ] is polynomial-time [Kaltofen 85, 91].

3 Overview of my work 2 Theorem. Factorization in K[x] is undecidable even if K is an effective field [van der Waerden 35, Fröhlich and Shepherdson 55]. Theorem. Factorization in Z p [x] [Berlekamp 67] and Q[x] [LLL] is polynomial-time. If factorization in K[x] is polynomial-time then factorization in K[x 1,...,x n ] is polynomial-time [Kaltofen 82]. If arithmetic in K is polynomial-time then factorization in K[x 1,...,x n ] is polynomial-time [Kaltofen 85, 91]. Best arithm. complexity: Lecerf 06 d 3 with LinBox linear algebra

4 Division-free straight-line program example υ 1 c 1 x 1 ; υ 2 y c 2 ; Comment: c 1,c 2 are constants in K υ 3 υ 2 υ 2 ; υ 4 υ 3 + υ 1 ; υ 5 υ 4 x 3 ;. υ 101 υ υ 51 ; 3 The variable υ 101 holds a polynomial in K[x 1,x 2,...] Straight-line programs [Kaltofen 85] and black box programs [Kaltofen & Trager 88] for irreducible factors can be computed in random polynomial time in the input size and total degree.

5 Division-free straight-line program example υ 1 c 1 x 1 ; υ 2 y c 2 ; Comment: c 1,c 2 are constants in K υ 3 υ 2 υ 2 ; υ 4 υ 3 + υ 1 ; υ 5 υ 4 x 3 ;. υ 101 υ υ 51 ; 4 The variable υ 101 holds a polynomial in K[x 1,x 2,...] Straight-line programs [Kaltofen 85] and black box programs [Kaltofen & Trager 88] for irreducible factors can be computed in random polynomial time in the input size and total degree. used by V. Kabernets [2003] for complexity lower bounds.

6 Subquadratic complexity 5 Theorem. We have two algorithms that factor in Z 2 [x] in O(n 1.81 ) bit complexity [Kaltofen & Shoup 95].

7 Subquadratic complexity 6 Theorem. We have two algorithms that factor in Z 2 [x] in O(n 1.81 ) bit complexity [Kaltofen & Shoup 95]. Unfortunately, remains best-known complexity today Note: no complexity model tricks (output size, field operation count, etc.) possible

8 Approximate multivariate factorization 7 Conclusion on my exact algorithm [JSC 1(1) 85] D. Izraelevitz at Massachusetts Institute of Technology has already implemented a version of algorithm 1 using complex floating point arithmetic. Early experiments indicate that the linear systems computed in step (L) tend to be numerically ill-conditioned. How to overcome this numerical problem is an important question which we will investigate.

9 Approximate multivariate factorization 8 Conclusion on my exact algorithm [JSC 1(1) 85] D. Izraelevitz at Massachusetts Institute of Technology has already implemented a version of algorithm 1 using complex floating point arithmetic. Early experiments indicate that the linear systems computed in step (L) tend to be numerically ill-conditioned. How to overcome this numerical problem is an important question which we will investigate. Gao, Kaltofen, May, Yang, Zhi 2004: practical algorithms to find the factorization of a nearby factorizable polynomial given any f especially noisy f : Given f = f 1 f s + f noise, we find f 1,... f s s.t. f 1 f s f 1 f s f noise even for large noise: f noise / f 10 3

10 Kaltofen & Koiran 06: supersparse (lacunary) polynomials f = i c i X αi K[X] where X αi = X α i,1 1 X α i,n n 9 Input: ϕ(ζ) Z[ζ] monic irred.; let K = Q[ζ]/(ϕ(ζ)) a supersparse f(x) = t i=1 c i X αi K[X] a factor degree bound d Output: a list of all irreducible factors of f over K of degree d and their multiplicities (which is t except for any X j ) Bit complexity is: ( size( f)+d + deg(ϕ)+log ϕ )O(n) (sparse factors) ( size( f)+d + deg(ϕ)+log ϕ )O(1) (blackbox factors) where size( f) = t (dense-size(c i )+ log 2 (α i,1 α i,n + 2) ) i=1

11 Black box polynomials 10 x 1,...,x n K f(x 1,...,x n ) K f K[x 1,...,x n ] K an arbitrary field, e.g., rationals, reals, complexes Perform polynmial algebra operations, e.g., factorization with n O(1) black box calls, n O(1) arithmetic operations in K and n O(1) randomly selected elements in K

12 Kaltofen and Trager (1988) efficiently construct the following efficient program: 11 p 1,..., p n K b 1,...,b n Precomputed data including e 1,...,e n. Program makes oracle calls : a 1,...,a n c 1,...,c n f(x 1,...,x n ) f(x 1,...,x n ). f(a 1,...,a n ) f(b 1,...,b n ) f(c 1,...,c n )... h 1 (p 1,..., p n ) h 2 (p 1,..., p n ). h r (p 1,..., p n ) f(x 1,...,x n ) f(x 1,...,x n ) = h 1 (x 1,...,x n ) e 1 h r (x 1,...,x n ) e r h i K[x 1,...,x n ] irreducible.

13 Characterization of Factor Evaluation Program 12 Always evaluates the same associate of each factor x y vs. ( 1 x) (2y) 2 Construction of program is Monte-Carlo (might produce incorrect program with probability ε), and requires a factorization procedure for K[y], but the program itself is deterministic Program contains positive integer constants of value bounded by 2 deg( f)1+o(1) /ε Program makes O(deg( f) 2 ) oracle calls, none of whose inputs depends on another one s output, parallel version Furthermore, program performs deg( f) 2+o(1) arithmetic operations in K

14 Given a black box p 1,..., p n K f(p 1,..., p n ) K 13 f(x 1,...,x n ) K[x 1,...,x n ] K a field compute by multiple evaluation of this black box the sparse representation of f t f(x 1,...,x n ) = a i x e i,1 1 x e i,n n, a i 0 i=1 Several solutions that are polynomial-time in n and t: Zippel (1979, 1988), Ben-Or, Tiwari (1988) Kaltofen, Lakshman (1988) Grigoriev, Karpinski, Singer (1988) Mansour (1992) Kaltofen and Lee (2000)

15 Homotopy Method for Solving F(X) = 0 14 Known: Wanted: Solution to Solution to G(X) = 0 F(X) = 0 x 1 (0) x 1 (1) x 2 (0) x 2 (1) x 3 (0) x 3 (1).. x n (0) x n (1) Follow from y = 0 to y = 1 the solutions of H(X(y)) = (1 y)g(x(y))+yf(x(y))

16 Our Homotopy 15 For f(x 1,...,x n ) K[x 1,...,x n ] consider f(x,y) = f(x + b 1,Y(p 2 a 2 (p 1 b 1 ) b 2 )+a 2 X + b 2,...,Y(p n a n (p 1 b 1 ) b n )+a n X + b n ) The field elements a 2,...,a n,b 1,...,b n are pre-chosen ( known ) The field elements p 1,..., p n are input Notice: The polynomial f(x,0) is independent of p 1,..., p n and can be factored into r f(x,0) = g i (X) e i, g i (X) K[X] irreducible i=1

17 Our Homotopy 16 For f(x 1,...,x n ) K[x 1,...,x n ] consider f(x,y) = f(x + b 1,Y(p 2 a 2 (p 1 b 1 ) b 2 )+a 2 X + b 2,...,Y(p n a n (p 1 b 1 ) b n )+a n X + b n ) The field elements a 2,...,a n,b 1,...,b n are pre-chosen ( known ) The field elements p 1,..., p n are input Notice: The polynomial f(x,0) is independent of p 1,..., p n and can be factored into r f(x,0) = g i (X) e i, g i (X) K[X] irreducible i=1 By an effective Hilbert Irreducibility Theorem one can guarantee that the g i are distinct images of the factors of f g i (X) = h i (X +b 1,...,a n X +b n ), f(x 1,...,x n ) = enters randomization r h(x 1,...,x n ) e i i=1

18 Our Homotopy 17 For f(x 1,...,x n ) K[x 1,...,x n ] consider f(x,y) = f(x + b 1,Y(p 2 a 2 (p 1 b 1 ) b 2 )+a 2 X + b 2,...,Y(p n a n (p 1 b 1 ) b n )+a n X + b n ) The field elements a 2,...,a n,b 1,...,b n are pre-chosen ( known ) The field elements p 1,..., p n are input Notice: The polynomial f(x,0) is independent of p 1,..., p n and can be factored into r f(x,0) = g i (X) e i, g i (X) K[X] irreducible i=1 By Hensel Lifting we can follow the factorization to f(x,y) = r i=1 h i (X,Y) e i Now f(p 1 b 1,1) = f(p 1,..., p n ), i : h i (p 1 b 1,1) = h i (p 1,..., p n )

19 Four Corollaries 18 Corollary 1: (Parallel Factorization) For K = Q, we can compute in Monte Carlo N C all sparse factors of f of fixed degree and with no more than a given number t terms Corollary 2: (Sparse Rational Interpolation) Given a degree bound b max(deg( f),deg(g)) and a bound t for the maximum number of non-zero terms in both f and g, we can in Las Vegas polynomial-time in b and t compute from a black box for f/g the sparse representations of f and g

20 Four Corollaries 19 Corollary 1: (Parallel Factorization) For K = Q, we can compute in Monte Carlo N C all sparse factors of f of fixed degree and with no more than a given number t terms Corollary 2 [Kaltofen & Yang 07]: (Sparse Rational Interpol.) Given a degree bound b max(deg( f),deg(g)) we can in Monte Carlo polynomial-time in b and t f,t g (number of terms in f and g) compute the sparse representations of f, g.

21 Four Corollaries 20 Corollary 1: (Parallel Factorization) For K = Q, we can compute in Monte Carlo N C all sparse factors of f of fixed degree and with no more than a given number t terms Corollary 2 [Kaltofen & Yang 07]: (Sparse Rational Interpol.) Given a degree bound b max(deg( f),deg(g)) we can in Monte Carlo polynomial-time in b and t f,t g (number of terms in f and g) compute the sparse representations of f, g. Uses early termination [Kaltofen & Lee 03]; our algorithm is practical. Hybrid version based on [Giesbrecht, Labahn, Lee 06] and [Kaltofen, Yang, Zhi 05].

22 Corollary 3: (Greatest Common Divisor) From a black box for f 1 (x 1,...,x n ),..., f r (x 1,...,x r ) K[x 1,...,x n ] we can efficiently produce a feasible program with oracle calls that allows to evaluate one and the same associate of GCD( f 1,..., f r ). 21

23 Corollary 4: (Factors as Straight-Line Programs) Let f K[x 1,...,x n ] be given by a straight-line program of size s, e.g., υ 1 c 1 x 1 ; υ 2 x 2 c 2 ; Comment: c 1,c 2 are constants in K υ 3 υ 2 υ 2 ; υ 4 υ 3 + υ 1 ; υ 5 υ 4 x 3 ;. υ 101 υ υ 51 ; 22 The variable υ 101 holds a polynomial in F q [x 1,...] of degree Then one can compute in polynomial-time in s+deg( f) straight-line programs of polynomial-size for all irreducible factors.

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