Factoring bivariate lacunary polynomials without heights Bruno - PowerPoint PPT Presentation

Factoring bivariate lacunary polynomials without heights Bruno Grenet ´ ENS Lyon Joint work with Arkadev Chattophyay Pascal Koiran ´ TIFR, Mumbai ENS Lyon Natacha Portier Yann Strozecki ´ ENS Lyon U. Versailles Palindromic Dagstuhl Seminar 13031: Computational Counting January 16. 2013

Representation of Univariate Polynomials P ( X ) = X 10 − 4 X 8 + 8 X 7 + 5 X 3 + 1 Representations ◮ Dense: [1 , 0 , − 4 , 8 , 0 , 0 , 0 , 5 , 0 , 0 , 1] ◮ Sparse: � � (10 : 1) , (8 : − 4) , (7 : 8) , (3 : 5) , (0 : 1) Factoring bivariate lacunary polynomials without heights 2 / 22 �

Representation of Multivariate Polynomials P ( X , Y , Z ) = X 2 Y 3 Z 5 − 4 X 3 Y 3 Z 2 + 8 X 5 Z 2 + 5 XYZ + 1 Representations ◮ Dense: [1 , . . . , − 4 , . . . , 8 , . . . , 5 , . . . , 1] ◮ Lacunary (supersparse): � � (2 , 3 , 5 : 1) , (3 , 3 , 2 : − 4) , (5 , 0 , 2 : 8) , (1 , 1 , 1 : 5) , (0 : 1) Factoring bivariate lacunary polynomials without heights 2 / 22 �

Size of the lacunary representation Definition k � a j X α 1 j · · · X α nj P ( X 1 , . . . , X n ) = n 1 j =1 k � = ⇒ size( P ) = size( a j ) + log( α 1 j ) + · · · + log( α nj ) j =1 Factoring bivariate lacunary polynomials without heights 3 / 22 �

Factorization of lacunary polynomials − X 6 − X 2 Y + X 5 Y + XY 2 − X 4 YZ − Y 2 Z + X 4 Z 2 + YZ 2 Factoring bivariate lacunary polynomials without heights 4 / 22 �

Factorization of lacunary polynomials − X 6 − X 2 Y + X 5 Y + XY 2 − X 4 YZ − Y 2 Z + X 4 Z 2 + YZ 2 = ( X − Y + Z )( X 4 + Y )( Z − X ) Factoring bivariate lacunary polynomials without heights 4 / 22 �

Factorization of lacunary polynomials − X 6 − X 2 Y + X 5 Y + XY 2 − X 4 YZ − Y 2 Z + X 4 Z 2 + YZ 2 = ( X − Y + Z )( X 4 + Y )( Z − X ) Factorization of a polynomial P Find F 1 , . . . , F t s.t. P = F 1 × · · · × F t Factoring bivariate lacunary polynomials without heights 4 / 22 �

Factorization of lacunary polynomials − X 6 − X 2 Y + X 5 Y + XY 2 − X 4 YZ − Y 2 Z + X 4 Z 2 + YZ 2 = ( X − Y + Z )( X 4 + Y )( Z − X ) Factorization of a polynomial P Find F 1 , . . . , F t s.t. P = F 1 × · · · × F t Proposition A lacunary polynomial can have exponentially many factors . = ⇒ restriction to finding some factors Factoring bivariate lacunary polynomials without heights 4 / 22 �

Factorization of sparse univariate polynomials k k � � a j X α j P ( X ) = size( P ) = size( a j ) + log( α j ) j =1 j =1 Factoring bivariate lacunary polynomials without heights 5 / 22 �

Factorization of sparse univariate polynomials k k � � a j X α j P ( X ) = size( P ) = size( a j ) + log( α j ) j =1 j =1 Theorem (Cucker-Koiran-Smale’98) Polynomial-time algorithm to find integer roots if a j ∈ Z . Factoring bivariate lacunary polynomials without heights 5 / 22 �

Factorization of sparse univariate polynomials k k � � a j X α j P ( X ) = size( P ) = size( a j ) + log( α j ) j =1 j =1 Theorem (Cucker-Koiran-Smale’98) Polynomial-time algorithm to find integer roots if a j ∈ Z . Theorem (Lenstra’99) Polynomial-time algorithm to find factors of degree ≤ d if a j ∈ K , where K is an algebraic number field. Factoring bivariate lacunary polynomials without heights 5 / 22 �

Factorization of lacunary polynomials Theorem (Kaltofen-Koiran’05) Polynomial-time algorithm to find linear factors of bivariate la- cunary polynomials over Q . Factoring bivariate lacunary polynomials without heights 6 / 22 �

Factorization of lacunary polynomials Theorem (Kaltofen-Koiran’05) Polynomial-time algorithm to find linear factors of bivariate la- cunary polynomials over Q . Theorem (Kaltofen-Koiran’06) Polynomial-time algorithm to find low-degree factors of multi- variate lacunary polynomials over algebraic number fields. Factoring bivariate lacunary polynomials without heights 6 / 22 �

Common ideas Gap Theorem ℓ k � � a j X α j Y β j a j X α j Y β j P = + j =1 j = ℓ +1 � �� � � �� � P 0 P 1 with α 1 ≤ α 2 ≤ · · · ≤ α k . Factoring bivariate lacunary polynomials without heights 7 / 22 �

Common ideas Gap Theorem ℓ k � � a j X α j Y β j a j X α j Y β j P = + j =1 j = ℓ +1 � �� � � �� � P 0 P 1 with α 1 ≤ α 2 ≤ · · · ≤ α k . Suppose that α ℓ +1 − α ℓ > gap( P ) Factoring bivariate lacunary polynomials without heights 7 / 22 �

Common ideas Gap Theorem ℓ k � � a j X α j Y β j a j X α j Y β j P = + j =1 j = ℓ +1 � �� � � �� � P 0 P 1 with α 1 ≤ α 2 ≤ · · · ≤ α k . Suppose that α ℓ +1 − α ℓ > gap( P ) , then F divides P iff F divides both P 0 and P 1 . Factoring bivariate lacunary polynomials without heights 7 / 22 �

Common ideas Gap Theorem ℓ k � � a j X α j Y β j a j X α j Y β j P = + j =1 j = ℓ +1 � �� � � �� � P 0 P 1 with α 1 ≤ α 2 ≤ · · · ≤ α k . Suppose that α ℓ +1 − α ℓ > gap( P ) , then F divides P iff F divides both P 0 and P 1 . gap( P ) : function of the algebraic height of P . Factoring bivariate lacunary polynomials without heights 7 / 22 �

Common algorithmic idea ◮ Recursively apply the Gap Theorem: P = X α 1 P 1 + · · · + X α t P s with deg( P t ) ≤ gap( P ) Factoring bivariate lacunary polynomials without heights 8 / 22 �

Common algorithmic idea ◮ Recursively apply the Gap Theorem: P = X α 1 P 1 + · · · + X α t P s with deg( P t ) ≤ gap( P ) ◮ Factor out P 1 , . . . , P t using a dense factorization algorithm Factoring bivariate lacunary polynomials without heights 8 / 22 �

Common algorithmic idea ◮ Recursively apply the Gap Theorem: P = X α 1 P 1 + · · · + X α t P s with deg( P t ) ≤ gap( P ) ◮ Factor out P 1 , . . . , P t using a dense factorization algorithm ◮ Refinements: Factoring bivariate lacunary polynomials without heights 8 / 22 �

Common algorithmic idea ◮ Recursively apply the Gap Theorem: P = X α 1 P 1 + · · · + X α t P s with deg( P t ) ≤ gap( P ) ◮ Factor out P 1 , . . . , P t using a dense factorization algorithm ◮ Refinements: • Factor out the gcd of the P t ’s Factoring bivariate lacunary polynomials without heights 8 / 22 �

Common algorithmic idea ◮ Recursively apply the Gap Theorem: P = X α 1 P 1 + · · · + X α t P s with deg( P t ) ≤ gap( P ) ◮ Factor out P 1 , . . . , P t using a dense factorization algorithm ◮ Refinements: • Factor out the gcd of the P t ’s • Factor out only one P t & check which factors are common to the other P t ’s Factoring bivariate lacunary polynomials without heights 8 / 22 �

Common algorithmic idea ◮ Recursively apply the Gap Theorem: P = X α 1 P 1 + · · · + X α t P s with deg( P t ) ≤ gap( P ) ◮ Factor out P 1 , . . . , P t using a dense factorization algorithm ◮ Refinements: • Factor out the gcd of the P t ’s • Factor out only one P t & check which factors are common to the other P t ’s • . . . Factoring bivariate lacunary polynomials without heights 8 / 22 �

Results Theorem Polynomial time algorithm to find multilinear factors of bivariate lacunary polynomials over algebraic number fields. Factoring bivariate lacunary polynomials without heights 9 / 22 �

Results Theorem Polynomial time algorithm to find multilinear factors of bivariate lacunary polynomials over algebraic number fields. ◮ Linear factors of bivariate lacunary polynomials [Kaltofen-Koiran’05] Factoring bivariate lacunary polynomials without heights 9 / 22 �

Results Theorem Polynomial time algorithm to find multilinear factors of bivariate lacunary polynomials over algebraic number fields. ◮ Linear factors of bivariate lacunary polynomials [Kaltofen-Koiran’05] ◮ gap( P ) independent of the height Factoring bivariate lacunary polynomials without heights 9 / 22 �

Results Theorem Polynomial time algorithm to find multilinear factors of bivariate lacunary polynomials over algebraic number fields. ◮ Linear factors of bivariate lacunary polynomials [Kaltofen-Koiran’05] ◮ gap( P ) independent of the height � More elementary algorithms Factoring bivariate lacunary polynomials without heights 9 / 22 �

Results Theorem Polynomial time algorithm to find multilinear factors of bivariate lacunary polynomials over algebraic number fields. ◮ Linear factors of bivariate lacunary polynomials [Kaltofen-Koiran’05] ◮ gap( P ) independent of the height � More elementary algorithms � Gap Theorem valid over any field of characteristic 0 Factoring bivariate lacunary polynomials without heights 9 / 22 �

Results Theorem Polynomial time algorithm to find multilinear factors of bivariate lacunary polynomials over algebraic number fields. ◮ Linear factors of bivariate lacunary polynomials [Kaltofen-Koiran’05] ◮ gap( P ) independent of the height � More elementary algorithms � Gap Theorem valid over any field of characteristic 0 ◮ Extension to multilinear factors Factoring bivariate lacunary polynomials without heights 9 / 22 �