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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

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Factoring bivariate lacunary polynomials without heights

Bruno Grenet

´ ENS Lyon Joint work with

Arkadev Chattophyay

TIFR, Mumbai

Pascal Koiran

´ ENS Lyon

Natacha Portier

´ ENS Lyon

Yann Strozecki

  • U. Versailles

Palindromic Dagstuhl Seminar 13031: Computational Counting January 16. 2013

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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)
  • 2 / 22

Factoring bivariate lacunary polynomials without heights

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Representation of Multivariate Polynomials

P(X, Y , Z) = X 2Y 3Z 5 − 4 X 3Y 3Z 2 + 8 X 5Z 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)
  • 2 / 22

Factoring bivariate lacunary polynomials without heights

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Size of the lacunary representation

Definition P(X1, . . . , Xn) =

k

  • j=1

ajX α1j

1

· · · X αnj

n

= ⇒ size(P) =

k

  • j=1

size(aj) + log(α1j) + · · · + log(αnj)

3 / 22 Factoring bivariate lacunary polynomials without heights

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Factorization of lacunary polynomials

−X 6 − X 2Y + X 5Y + XY 2 − X 4YZ − Y 2Z + X 4Z 2 + YZ 2

4 / 22 Factoring bivariate lacunary polynomials without heights

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Factorization of lacunary polynomials

−X 6 − X 2Y + X 5Y + XY 2 − X 4YZ − Y 2Z + X 4Z 2 + YZ 2 = (X − Y + Z)(X 4 + Y )(Z − X)

4 / 22 Factoring bivariate lacunary polynomials without heights

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Factorization of lacunary polynomials

−X 6 − X 2Y + X 5Y + XY 2 − X 4YZ − Y 2Z + X 4Z 2 + YZ 2 = (X − Y + Z)(X 4 + Y )(Z − X) Factorization of a polynomial P Find F1, . . . , Ft s.t. P = F1 × · · · × Ft

4 / 22 Factoring bivariate lacunary polynomials without heights

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Factorization of lacunary polynomials

−X 6 − X 2Y + X 5Y + XY 2 − X 4YZ − Y 2Z + X 4Z 2 + YZ 2 = (X − Y + Z)(X 4 + Y )(Z − X) Factorization of a polynomial P Find F1, . . . , Ft s.t. P = F1 × · · · × Ft Proposition A lacunary polynomial can have exponentially many factors. = ⇒ restriction to finding some factors

4 / 22 Factoring bivariate lacunary polynomials without heights

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Factorization of sparse univariate polynomials

P(X) =

k

  • j=1

ajX αj size(P) =

k

  • j=1

size(aj) + log(αj)

5 / 22 Factoring bivariate lacunary polynomials without heights

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Factorization of sparse univariate polynomials

P(X) =

k

  • j=1

ajX αj size(P) =

k

  • j=1

size(aj) + log(αj) Theorem (Cucker-Koiran-Smale’98) Polynomial-time algorithm to find integer roots if aj ∈ Z.

5 / 22 Factoring bivariate lacunary polynomials without heights

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Factorization of sparse univariate polynomials

P(X) =

k

  • j=1

ajX αj size(P) =

k

  • j=1

size(aj) + log(αj) Theorem (Cucker-Koiran-Smale’98) Polynomial-time algorithm to find integer roots if aj ∈ Z. Theorem (Lenstra’99) Polynomial-time algorithm to find factors of degree ≤ d if aj ∈ K, where K is an algebraic number field.

5 / 22 Factoring bivariate lacunary polynomials without heights

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Factorization of lacunary polynomials

Theorem (Kaltofen-Koiran’05) Polynomial-time algorithm to find linear factors of bivariate la- cunary polynomials over Q.

6 / 22 Factoring bivariate lacunary polynomials without heights

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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.

6 / 22 Factoring bivariate lacunary polynomials without heights

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Common ideas

Gap Theorem P =

ℓ

  • j=1

ajX αjY βj

  • P0

+

k

  • j=ℓ+1

ajX αjY βj

  • P1

with α1 ≤ α2 ≤ · · · ≤ αk.

7 / 22 Factoring bivariate lacunary polynomials without heights

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Common ideas

Gap Theorem P =

ℓ

  • j=1

ajX αjY βj

  • P0

+

k

  • j=ℓ+1

ajX αjY βj

  • P1

with α1 ≤ α2 ≤ · · · ≤ αk. Suppose that αℓ+1 − αℓ > gap(P)

7 / 22 Factoring bivariate lacunary polynomials without heights

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Common ideas

Gap Theorem P =

ℓ

  • j=1

ajX αjY βj

  • P0

+

k

  • j=ℓ+1

ajX αjY βj

  • P1

with α1 ≤ α2 ≤ · · · ≤ αk. Suppose that αℓ+1 − αℓ > gap(P), then F divides P iff F divides both P0 and P1.

7 / 22 Factoring bivariate lacunary polynomials without heights

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Common ideas

Gap Theorem P =

ℓ

  • j=1

ajX αjY βj

  • P0

+

k

  • j=ℓ+1

ajX αjY βj

  • P1

with α1 ≤ α2 ≤ · · · ≤ αk. Suppose that αℓ+1 − αℓ > gap(P), then F divides P iff F divides both P0 and P1. gap(P): function of the algebraic height of P.

7 / 22 Factoring bivariate lacunary polynomials without heights

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Common algorithmic idea

◮ Recursively apply the Gap Theorem:

P = X α1P1 + · · · + X αtPs with deg(Pt) ≤ gap(P)

8 / 22 Factoring bivariate lacunary polynomials without heights

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Common algorithmic idea

◮ Recursively apply the Gap Theorem:

P = X α1P1 + · · · + X αtPs with deg(Pt) ≤ gap(P)

◮ Factor out P1, . . . , Pt using a dense factorization algorithm

8 / 22 Factoring bivariate lacunary polynomials without heights

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Common algorithmic idea

◮ Recursively apply the Gap Theorem:

P = X α1P1 + · · · + X αtPs with deg(Pt) ≤ gap(P)

◮ Factor out P1, . . . , Pt using a dense factorization algorithm ◮ Refinements:

8 / 22 Factoring bivariate lacunary polynomials without heights

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Common algorithmic idea

◮ Recursively apply the Gap Theorem:

P = X α1P1 + · · · + X αtPs with deg(Pt) ≤ gap(P)

◮ Factor out P1, . . . , Pt using a dense factorization algorithm ◮ Refinements:

  • Factor out the gcd of the Pt’s

8 / 22 Factoring bivariate lacunary polynomials without heights

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Common algorithmic idea

◮ Recursively apply the Gap Theorem:

P = X α1P1 + · · · + X αtPs with deg(Pt) ≤ gap(P)

◮ Factor out P1, . . . , Pt using a dense factorization algorithm ◮ Refinements:

  • Factor out the gcd of the Pt’s
  • Factor out only one Pt & check which factors are common to the
  • ther Pt’s

8 / 22 Factoring bivariate lacunary polynomials without heights

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Common algorithmic idea

◮ Recursively apply the Gap Theorem:

P = X α1P1 + · · · + X αtPs with deg(Pt) ≤ gap(P)

◮ Factor out P1, . . . , Pt using a dense factorization algorithm ◮ Refinements:

  • Factor out the gcd of the Pt’s
  • Factor out only one Pt & check which factors are common to the
  • ther Pt’s
  • . . .

8 / 22 Factoring bivariate lacunary polynomials without heights

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Results

Theorem Polynomial time algorithm to find multilinear factors of bivariate lacunary polynomials over algebraic number fields.

9 / 22 Factoring bivariate lacunary polynomials without heights

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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]

9 / 22 Factoring bivariate lacunary polynomials without heights

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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

9 / 22 Factoring bivariate lacunary polynomials without heights

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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

9 / 22 Factoring bivariate lacunary polynomials without heights

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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

9 / 22 Factoring bivariate lacunary polynomials without heights

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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

9 / 22 Factoring bivariate lacunary polynomials without heights

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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 ◮ Results in positive characteristics

9 / 22 Factoring bivariate lacunary polynomials without heights

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Linear factors of bivariate polynomials

P(X, Y ) =

k

  • j=1

ajX αjY βj

10 / 22 Factoring bivariate lacunary polynomials without heights

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Linear factors of bivariate polynomials

P(X, Y ) =

k

  • j=1

ajX αjY βj Observation (Y − uX − v) divides P(X, Y ) ⇐ ⇒ P(X, uX + v) ≡ 0

10 / 22 Factoring bivariate lacunary polynomials without heights

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Linear factors of bivariate polynomials

P(X, Y ) =

k

  • j=1

ajX αjY βj Observation (Y − uX − v) divides P(X, Y ) ⇐ ⇒ P(X, uX + v) ≡ 0

◮ Study of polynomials of the form

  • j

ajX αj(uX + v)βj

10 / 22 Factoring bivariate lacunary polynomials without heights

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Linear factors of bivariate polynomials

P(X, Y ) =

k

  • j=1

ajX αjY βj Observation (Y − uX − v) divides P(X, Y ) ⇐ ⇒ P(X, uX + v) ≡ 0

◮ Study of polynomials of the form

  • j

ajX αj(uX + v)βj

◮ K: any field of characteristic 0

10 / 22 Factoring bivariate lacunary polynomials without heights

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Bound on the valuation

Definition val(P) = degree of the lowest degree monomial of P ∈ K[X]

11 / 22 Factoring bivariate lacunary polynomials without heights

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Bound on the valuation

Definition val(P) = degree of the lowest degree monomial of P ∈ K[X] Theorem P =

k

  • j=1

ajX αj(uX + v)βj ≡ 0, with α1 ≤ · · · ≤ αk

11 / 22 Factoring bivariate lacunary polynomials without heights

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Bound on the valuation

Definition val(P) = degree of the lowest degree monomial of P ∈ K[X] Theorem P =

k

  • j=1

ajX αj(uX + v)βj ≡ 0, with α1 ≤ · · · ≤ αk = ⇒ val(P) ≤ max

1≤j≤k

  • αj +

k + 1 − j 2

  • 11 / 22

Factoring bivariate lacunary polynomials without heights

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Bound on the valuation

Definition val(P) = degree of the lowest degree monomial of P ∈ K[X] Theorem P =

k

  • j=1

ajX αj(uX + v)βj ≡ 0, with α1 ≤ · · · ≤ αk = ⇒ val(P) ≤ α1 + k 2

  • ◮ X αj(uX + v)βj linearly independent

11 / 22 Factoring bivariate lacunary polynomials without heights

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Bound on the valuation

Definition val(P) = degree of the lowest degree monomial of P ∈ K[X] Theorem P =

k

  • j=1

ajX αj(uX + v)βj ≡ 0, with α1 ≤ · · · ≤ αk = ⇒ val(P) ≤ α1 + k 2

  • ◮ X αj(uX + v)βj linearly independent

◮ Haj´

  • s’ Lemma: if α1 = · · · = αk, val(P) ≤ α1 + (k − 1)

11 / 22 Factoring bivariate lacunary polynomials without heights

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The Wronskian

Definition Let f1, . . . , fk ∈ K[X]. Then wr(f1, . . . , fk) = det      f1 f2 . . . fk f ′

1

f ′

2

. . . f ′

k

. . . . . . . . . f (k−1)

1

f (k−1)

2

. . . f (k−1)

k

     .

12 / 22 Factoring bivariate lacunary polynomials without heights

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The Wronskian

Definition Let f1, . . . , fk ∈ K[X]. Then wr(f1, . . . , fk) = det      f1 f2 . . . fk f ′

1

f ′

2

. . . f ′

k

. . . . . . . . . f (k−1)

1

f (k−1)

2

. . . f (k−1)

k

     . Proposition (Bˆ

  • cher, 1900)

wr(f1, . . . , fk) = 0 ⇐ ⇒ the fj’s are linearly independent.

12 / 22 Factoring bivariate lacunary polynomials without heights

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Wronskian & valuation

Lemma val(wr(f1, . . . , fk)) ≥

k

  • j=1

val(fj) − k 2

  • 13 / 22

Factoring bivariate lacunary polynomials without heights

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Wronskian & valuation

Lemma val(wr(f1, . . . , fk)) ≥

k

  • j=1

val(fj) − k 2

  • Lemma

Let fj = X αj(uX + v)βj, linearly independent, s.t. αj, βj ≥ k − 1. val(wr(f1, . . . , fk)) ≤

k

  • j=1

αj

13 / 22 Factoring bivariate lacunary polynomials without heights

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Wronskian & valuation

Lemma val(wr(f1, . . . , fk)) ≥

k

  • j=1

val(fj) − k 2

  • Lemma

Let fj = X αj(uX + v)βj, linearly independent, s.t. αj, βj ≥ k − 1. val(wr(f1, . . . , fk)) ≤

k

  • j=1

αj Proof of the theorem. wr(P, f2, . . . , fk) = a1 wr(f1, . . . , fk)

13 / 22 Factoring bivariate lacunary polynomials without heights

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Wronskian & valuation

Lemma val(wr(f1, . . . , fk)) ≥

k

  • j=1

val(fj) − k 2

  • Lemma

Let fj = X αj(uX + v)βj, linearly independent, s.t. αj, βj ≥ k − 1. val(wr(f1, . . . , fk)) ≤

k

  • j=1

αj Proof of the theorem. wr(P, f2, . . . , fk) = a1 wr(f1, . . . , fk)

k

  • j=1

αj ≥ val(wr(f1, . . . , fk)) ≥ val(P) +

k

  • j=2

αj − k 2

  • 13 / 22

Factoring bivariate lacunary polynomials without heights

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With some dirty computations. . .

◮ Generalization:

k

  • j=1

aj

m

  • i=1

f αij

i

, deg(fi) ≤ d

14 / 22 Factoring bivariate lacunary polynomials without heights

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With some dirty computations. . .

◮ Generalization:

k

  • j=1

aj

m

  • i=1

f αij

i

, deg(fi) ≤ d

◮ Lower bound: ∃P, val(P) ≥ α1 + (2k − 3)

14 / 22 Factoring bivariate lacunary polynomials without heights

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With some dirty computations. . .

◮ Generalization:

k

  • j=1

aj

m

  • i=1

f αij

i

, deg(fi) ≤ d

◮ Lower bound: ∃P, val(P) ≥ α1 + (2k − 3)

−1 + (1 + X)2k+3 −

k

  • j=0

2k + 3 2j + 1 k + 1 + j k + 1 − j

  • X 2j+1(1 + X)k+1−j

14 / 22 Factoring bivariate lacunary polynomials without heights

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With some dirty computations. . .

◮ Generalization:

k

  • j=1

aj

m

  • i=1

f αij

i

, deg(fi) ≤ d

◮ Lower bound: ∃P, val(P) ≥ α1 + (2k − 3)

−1 + (1 + X)2k+3 −

k

  • j=0

2k + 3 2j + 1 k + 1 + j k + 1 − j

  • X 2j+1(1 + X)k+1−j

= X 2k+3

14 / 22 Factoring bivariate lacunary polynomials without heights

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Gap Theorem

Theorem Let P =

ℓ

  • j=1

ajX αj(uX + v)βj

  • P0

+

k

  • j=ℓ+1

ajX αj(uX + v)βj

  • P1

with u, v = 0, α1 ≤ · · · ≤ αk. If αℓ+1 > max

1≤j≤ℓ

  • αj +

ℓ + 1 − j 2

  • ,

then P ≡ 0 iff both P0 ≡ 0 and P1 ≡ 0.

15 / 22 Factoring bivariate lacunary polynomials without heights

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Gap Theorem

Theorem Let P =

ℓ

  • j=1

ajX αj(uX + v)βj

  • P0

+

k

  • j=ℓ+1

ajX αj(uX + v)βj

  • P1

with u, v = 0, α1 ≤ · · · ≤ αk. If ℓ is the smallest index s.t. αℓ+1 > α1 + ℓ 2

  • ,

then P ≡ 0 iff both P0 ≡ 0 and P1 ≡ 0.

15 / 22 Factoring bivariate lacunary polynomials without heights

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Finding linear factors

Observation + Gap Theorem (Y − uX − v) divides P(X, Y ) ⇐ ⇒ P(X, uX + v) ≡ 0

16 / 22 Factoring bivariate lacunary polynomials without heights

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Finding linear factors

Observation + Gap Theorem (Y − uX − v) divides P(X, Y ) ⇐ ⇒ P(X, uX + v) ≡ 0 ⇐ ⇒ P1(X, uX + v) ≡ · · · ≡ Ps(X, uX + v) ≡ 0

16 / 22 Factoring bivariate lacunary polynomials without heights

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Finding linear factors

Observation + Gap Theorem (Y − uX − v) divides P(X, Y ) ⇐ ⇒ P(X, uX + v) ≡ 0 ⇐ ⇒ P1(X, uX + v) ≡ · · · ≡ Ps(X, uX + v) ≡ 0 ⇐ ⇒ (Y − uX − v) divides each Pt(X, Y )

16 / 22 Factoring bivariate lacunary polynomials without heights

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Finding linear factors

Observation + Gap Theorem (Y − uX − v) divides P(X, Y ) ⇐ ⇒ P(X, uX + v) ≡ 0 ⇐ ⇒ P1(X, uX + v) ≡ · · · ≡ Ps(X, uX + v) ≡ 0 ⇐ ⇒ (Y − uX − v) divides each Pt(X, Y ) find linear factors of low-degree polynomials

16 / 22 Factoring bivariate lacunary polynomials without heights

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Finding linear factors

Observation + Gap Theorem (Y − uX − v) divides P(X, Y ) ⇐ ⇒ P(X, uX + v) ≡ 0 ⇐ ⇒ P1(X, uX + v) ≡ · · · ≡ Ps(X, uX + v) ≡ 0 ⇐ ⇒ (Y − uX − v) divides each Pt(X, Y ) find linear factors of low-degree polynomials Remark We need K to be an algebraic number field.

16 / 22 Factoring bivariate lacunary polynomials without heights

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Some details

Find linear factors (Y − uX − v) of P(X, Y ) =

k

  • j=1

ajX αjY βj

17 / 22 Factoring bivariate lacunary polynomials without heights

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SLIDE 58

Some details

Find linear factors (Y − uX − v) of P(X, Y ) =

k

  • j=1

ajX αjY βj

  • 1. If u = 0: Factors of polynomials

j ajY βj

17 / 22 Factoring bivariate lacunary polynomials without heights

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SLIDE 59

Some details

Find linear factors (Y − uX − v) of P(X, Y ) =

k

  • j=1

ajX αjY βj

  • 1. If u = 0: Factors of polynomials

j ajY βj

[Lenstra’99]

17 / 22 Factoring bivariate lacunary polynomials without heights

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SLIDE 60

Some details

Find linear factors (Y − uX − v) of P(X, Y ) =

k

  • j=1

ajX αjY βj

  • 1. If u = 0: Factors of polynomials

j ajY βj

[Lenstra’99]

  • 2. If v = 0: P(X, uX) =

j ajuβjX αj+βj

17 / 22 Factoring bivariate lacunary polynomials without heights

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SLIDE 61

Some details

Find linear factors (Y − uX − v) of P(X, Y ) =

k

  • j=1

ajX αjY βj

  • 1. If u = 0: Factors of polynomials

j ajY βj

[Lenstra’99]

  • 2. If v = 0: P(X, uX) =

j ajuβjX αj+βj

[Lenstra’99]

17 / 22 Factoring bivariate lacunary polynomials without heights

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SLIDE 62

Some details

Find linear factors (Y − uX − v) of P(X, Y ) =

k

  • j=1

ajX αjY βj

  • 1. If u = 0: Factors of polynomials

j ajY βj

[Lenstra’99]

  • 2. If v = 0: P(X, uX) =

j ajuβjX αj+βj

[Lenstra’99]

  • 3. If u, v = 0:

17 / 22 Factoring bivariate lacunary polynomials without heights

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SLIDE 63

Some details

Find linear factors (Y − uX − v) of P(X, Y ) =

k

  • j=1

ajX αjY βj

  • 1. If u = 0: Factors of polynomials

j ajY βj

[Lenstra’99]

  • 2. If v = 0: P(X, uX) =

j ajuβjX αj+βj

[Lenstra’99]

  • 3. If u, v = 0:
  • Compute P = P1 + · · · + Ps where Pt =

j ajX αj Y βj with

αmax ≤ αmin + k

2

  • 17 / 22

Factoring bivariate lacunary polynomials without heights

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Some details

Find linear factors (Y − uX − v) of P(X, Y ) =

k

  • j=1

ajX αjY βj

  • 1. If u = 0: Factors of polynomials

j ajY βj

[Lenstra’99]

  • 2. If v = 0: P(X, uX) =

j ajuβjX αj+βj

[Lenstra’99]

  • 3. If u, v = 0:
  • Compute P = P1 + · · · + Ps where Pt =

j ajX αj Y βj with

αmax ≤ αmin + k

2

  • Invert the roles of X and Y , to get βmax ≤ βmin +

k

2

  • 17 / 22

Factoring bivariate lacunary polynomials without heights

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Some details

Find linear factors (Y − uX − v) of P(X, Y ) =

k

  • j=1

ajX αjY βj

  • 1. If u = 0: Factors of polynomials

j ajY βj

[Lenstra’99]

  • 2. If v = 0: P(X, uX) =

j ajuβjX αj+βj

[Lenstra’99]

  • 3. If u, v = 0:
  • Compute P = P1 + · · · + Ps where Pt =

j ajX αj Y βj with

αmax ≤ αmin + k

2

  • Invert the roles of X and Y , to get βmax ≤ βmin +

k

2

  • Apply some dense factorization algorithm [Kaltofen’82, . . . , Lecerf’07]

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SLIDE 66

Complexity

Main computational task: Factorization of dense polynomials

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SLIDE 67

Complexity

Main computational task: Factorization of dense polynomials = ⇒ Complexity in terms of gap(P)

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SLIDE 68

Complexity

Main computational task: Factorization of dense polynomials = ⇒ Complexity in terms of gap(P)

◮ [Kaltofen-Koiran’05]: gap(P) = O(k log k + k log hP)

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SLIDE 69

Complexity

Main computational task: Factorization of dense polynomials = ⇒ Complexity in terms of gap(P)

◮ [Kaltofen-Koiran’05]: gap(P) = O(k log k + k log hP)

Ex.: hP = maxj |aj| if P ∈ Z[X, Y ]

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SLIDE 70

Complexity

Main computational task: Factorization of dense polynomials = ⇒ Complexity in terms of gap(P)

◮ [Kaltofen-Koiran’05]: gap(P) = O(k log k + k log hP) ◮ Here: gap(P) = O(k2)

Ex.: hP = maxj |aj| if P ∈ Z[X, Y ]

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SLIDE 71

Finding multilinear factors

Lemma Let P =

j ajX αj(uX + v)βj(wX + t)γj ≡ 0, uvwt = 0. Then

val(P) ≤ max

j

  • αj + 2

k + 1 − j 2

  • .

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SLIDE 72

Finding multilinear factors

Lemma Let P =

j ajX αj(uX + v)βj(wX + t)γj ≡ 0, uvwt = 0. Then

val(P) ≤ max

j

  • αj + 2

k + 1 − j 2

  • .

Theorem There exists a polynomial-time algorithm to compute the multi- linear factors of

j ajX αjY βj.

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SLIDE 73

Finding multilinear factors

Lemma Let P =

j ajX αj(uX + v)βj(wX + t)γj ≡ 0, uvwt = 0. Then

val(P) ≤ max

j

  • αj + 2

k + 1 − j 2

  • .

Theorem There exists a polynomial-time algorithm to compute the multi- linear factors of

j ajX αjY βj.

Proof.

◮ XY − (uX − vY + w) divides P ⇐

⇒ P(X, uX+w

X+v ) ≡ 0.

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SLIDE 74

Finding multilinear factors

Lemma Let P =

j ajX αj(uX + v)βj(wX + t)γj ≡ 0, uvwt = 0. Then

val(P) ≤ max

j

  • αj + 2

k + 1 − j 2

  • .

Theorem There exists a polynomial-time algorithm to compute the multi- linear factors of

j ajX αjY βj.

Proof.

◮ XY − (uX − vY + w) divides P ⇐

⇒ P(X, uX+w

X+v ) ≡ 0.

◮ Gap Theorem for Q(X) = (X + v)maxj βjP(X, uX+w

X+v ).

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SLIDE 75

Positive characteristic

(1 + X)2n + (1 + X)2n+1 = X 2n(X + 1)

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SLIDE 76

Positive characteristic

(1 + X)2n + (1 + X)2n+1 = X 2n(X + 1) Theorem Let P =

k

  • j=1

ajX αj(uX + v)βj ≡ 0, where p > maxj(αj + βj) and aj ∈ F

  • ps. Then val(P) ≤ maxj
  • αj +

k+1−j

2

  • .

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SLIDE 77

Algorithms in positive characteristic

Theorem Let P =

j ajX αjY βj ∈ F ps[X, Y ], where p > maxj(αj + βj).

Finding factors of the form (uX + vY + w) is

◮ doable in randomized polynomial time if uvw = 0 ;

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SLIDE 78

Algorithms in positive characteristic

Theorem Let P =

j ajX αjY βj ∈ F ps[X, Y ], where p > maxj(αj + βj).

Finding factors of the form (uX + vY + w) is

◮ doable in randomized polynomial time if uvw = 0 ; ◮ NP-hard under randomized reductions otherwise.

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SLIDE 79

Algorithms in positive characteristic

Theorem Let P =

j ajX αjY βj ∈ F ps[X, Y ], where p > maxj(αj + βj).

Finding factors of the form (uX + vY + w) is

◮ doable in randomized polynomial time if uvw = 0 ; ◮ NP-hard under randomized reductions otherwise. ◮ Only randomized dense factorization algorithms over Fps

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SLIDE 80

Algorithms in positive characteristic

Theorem Let P =

j ajX αjY βj ∈ F ps[X, Y ], where p > maxj(αj + βj).

Finding factors of the form (uX + vY + w) is

◮ doable in randomized polynomial time if uvw = 0 ; ◮ NP-hard under randomized reductions otherwise. ◮ Only randomized dense factorization algorithms over Fps ◮ NP-hardness: reduction from root detection over Fps

[Kipnis-Shamir’99, Bi-Cheng-Rojas’12]

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SLIDE 81

Conclusion

+ More elementary proofs for [Kaltofen & Koiran’05]

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SLIDE 82

Conclusion

+ More elementary proofs for [Kaltofen & Koiran’05] + Complexity independent of the height large coefficients

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SLIDE 83

Conclusion

+ More elementary proofs for [Kaltofen & Koiran’05] + Complexity independent of the height large coefficients + Consequences in large positive characteristic

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SLIDE 84

Conclusion

+ More elementary proofs for [Kaltofen & Koiran’05] + Complexity independent of the height large coefficients + Consequences in large positive characteristic − Still relies on [Lenstra’99] number fields

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SLIDE 85

Conclusion

+ More elementary proofs for [Kaltofen & Koiran’05] + Complexity independent of the height large coefficients + Consequences in large positive characteristic − Still relies on [Lenstra’99] number fields

◮ Can we find low-degree factors of multivariate polynomials?

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SLIDE 86

Conclusion

+ More elementary proofs for [Kaltofen & Koiran’05] + Complexity independent of the height large coefficients + Consequences in large positive characteristic − Still relies on [Lenstra’99] number fields

◮ Can we find low-degree factors of multivariate polynomials? ◮ And low-degree factors of univariate polynomials?

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SLIDE 87

Conclusion

+ More elementary proofs for [Kaltofen & Koiran’05] + Complexity independent of the height large coefficients + Consequences in large positive characteristic − Still relies on [Lenstra’99] number fields

◮ Can we find low-degree factors of multivariate polynomials? ◮ And low-degree factors of univariate polynomials?

Impossible in positive characteristic

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SLIDE 88

Conclusion

+ More elementary proofs for [Kaltofen & Koiran’05] + Complexity independent of the height large coefficients + Consequences in large positive characteristic − Still relies on [Lenstra’99] number fields

◮ Can we find low-degree factors of multivariate polynomials? ◮ And low-degree factors of univariate polynomials?

Impossible in positive characteristic

◮ Can we find lacunary factors?

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SLIDE 89

Conclusion

+ More elementary proofs for [Kaltofen & Koiran’05] + Complexity independent of the height large coefficients + Consequences in large positive characteristic − Still relies on [Lenstra’99] number fields

◮ Can we find low-degree factors of multivariate polynomials? ◮ And low-degree factors of univariate polynomials?

Impossible in positive characteristic

◮ Can we find lacunary factors? ◮ Can we handle polynomials in small characteristic?

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SLIDE 90

Conclusion

+ More elementary proofs for [Kaltofen & Koiran’05] + Complexity independent of the height large coefficients + Consequences in large positive characteristic − Still relies on [Lenstra’99] number fields

◮ Can we find low-degree factors of multivariate polynomials? ◮ And low-degree factors of univariate polynomials?

Impossible in positive characteristic

◮ Can we find lacunary factors? ◮ Can we handle polynomials in small characteristic? ◮ Is the correct bound for the valuation quadratic or linear?

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SLIDE 91

Conclusion

+ More elementary proofs for [Kaltofen & Koiran’05] + Complexity independent of the height large coefficients + Consequences in large positive characteristic − Still relies on [Lenstra’99] number fields

◮ Can we find low-degree factors of multivariate polynomials? ◮ And low-degree factors of univariate polynomials?

Impossible in positive characteristic

◮ Can we find lacunary factors? ◮ Can we handle polynomials in small characteristic? ◮ Is the correct bound for the valuation quadratic or linear? ◮ What is the complexity of counting the number of factors?

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SLIDE 92

Conclusion

+ More elementary proofs for [Kaltofen & Koiran’05] + Complexity independent of the height large coefficients + Consequences in large positive characteristic − Still relies on [Lenstra’99] number fields

◮ Can we find low-degree factors of multivariate polynomials? ◮ And low-degree factors of univariate polynomials?

Impossible in positive characteristic

◮ Can we find lacunary factors? ◮ Can we handle polynomials in small characteristic? ◮ Is the correct bound for the valuation quadratic or linear? ◮ What is the complexity of counting the number of factors?

arXiv:1206.4224

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