[PPT] - Computing low-degree factors of lacunary polynomials: a PowerPoint Presentation

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Computing low-degree factors of lacunary polynomials: a Newton-Puiseux Approach

Bruno Grenet LIRMM — Universit´ e Montpellier 2 JNCF — November 3., 2014

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Classical factorization algorithms

Factorization of a polynomial f Find f1, . . . , ft, irreducible, s.t. f = f1 × · · · × ft.

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Classical factorization algorithms

Factorization of a polynomial f Find f1, . . . , ft, irreducible, s.t. f = f1 × · · · × ft.

◮ Many algorithms

over Z, Q, Q(α), Q, Qp, F

q, R, C, . . . ;

in 1, 2, . . . , n variables.

◮ Complexity: polynomial in deg(f)

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Classical factorization algorithms

Factorization of a polynomial f Find f1, . . . , ft, irreducible, s.t. f = f1 × · · · × ft.

◮ Many algorithms

over Z, Q, Q(α), Q, Qp, F

q, R, C, . . . ;

in 1, 2, . . . , n variables.

◮ Complexity: polynomial in deg(f)

X102Y101 + X101Y102 − X101Y101 − X − Y + 1 = (X + Y − 1) × (X101Y101 − 1)

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Classical factorization algorithms

Factorization of a polynomial f Find f1, . . . , ft, irreducible, s.t. f = f1 × · · · × ft.

◮ Many algorithms

over Z, Q, Q(α), Q, Qp, F

q, R, C, . . . ;

in 1, 2, . . . , n variables.

◮ Complexity: polynomial in deg(f)

X102Y101 + X101Y102 − X101Y101 − X − Y + 1 = (X + Y − 1) × (X101Y101 − 1) = (X + Y − 1) × (XY − 1) × (1 + XY + · · · + X100Y100)

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Lacunary factorization algorithms

Definition

f(X1, . . . , Xn) =

k

j=1

cjXα1j

1

· · · Xαnj

n

◮ size(f) ≃ k

maxj(size(cj)) + nlog(deg f)
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Bruno Grenet – Computing low-degree factors of lacunary polynomials

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Lacunary factorization algorithms

Definition

f(X1, . . . , Xn) =

k

j=1

cjXα1j

1

· · · Xαnj

n

◮ size(f) ≃ k

maxj(size(cj)) + nlog(deg f)
Theorems

There exist deterministic polynomial-time algorithms computing

◮ linear factors (integer roots) of f ∈ Z[X]; [Cucker-Koiran-Smale’98] ◮ low-degree factors of f ∈ Q(α)[X]; [H. Lenstra’99] ◮ low-degree factors of f ∈ Q(α)[X1, . . . , Xn]. [Kaltofen-Koiran’06]

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Lacunary factorization algorithms

Definition

f(X1, . . . , Xn) =

k

j=1

cjXα1j

1

· · · Xαnj

n

◮ size(f) ≃ k

maxj(size(cj)) + nlog(deg f)
Theorems

There exist deterministic polynomial-time algorithms computing

◮ linear factors (integer roots) of f ∈ Z[X]; [Cucker-Koiran-Smale’98] ◮ low-degree factors of f ∈ Q(α)[X]; [H. Lenstra’99] ◮ low-degree factors of f ∈ Q(α)[X1, . . . , Xn]. [Kaltofen-Koiran’06]

It is NP-hard to compute roots of f ∈ F

p[X]. [Bi-Cheng-Rojas’13]

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Main result

Let K be any field of characteristic 0. Theorem The computation of the degree-d factors of f ∈ K[X1, . . . , Xn] reduces to

◮ univariate lacunary factorizations plus post-processing, and ◮ multivariate low-degree factorization,

in poly(size(f), d) bit operations.

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Main result

Let K be any field of characteristic 0. Theorem The computation of the degree-d factors of f ∈ K[X1, . . . , Xn] reduces to

◮ univariate lacunary factorizations plus post-processing, and ◮ multivariate low-degree factorization,

in poly(size(f), d) bit operations.

◮ New algorithm for K = Q(α); some factors for K = Q, R, C, Qp

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Main result

Let K be any field of characteristic 0. Theorem The computation of the degree-d factors of f ∈ K[X1, . . . , Xn] reduces to

◮ univariate lacunary factorizations plus post-processing, and ◮ multivariate low-degree factorization,

in poly(size(f), d) bit operations.

◮ New algorithm for K = Q(α); some factors for K = Q, R, C, Qp ◮ Case d = 1 [G.-Chattopadhyay-Koiran-Portier-Strozecki’13]

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Two kinds of factors

Definition A polynomial g =

j bjXγjYδj is (p, q)-homogeneous of order

ω if pγj + qδj = ω for all j. Otherwise, g is said inhomogeneous.

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Two kinds of factors

Definition A polynomial g =

j bjXγjYδj is (p, q)-homogeneous of order

ω if pγj + qδj = ω for all j. Otherwise, g is said inhomogeneous.

X Y 1 2 3 4 5 6 7 1 2 3 4 5 6

Univariate lacunary factorization

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Two kinds of factors

Definition A polynomial g =

j bjXγjYδj is (p, q)-homogeneous of order

ω if pγj + qδj = ω for all j. Otherwise, g is said inhomogeneous.

X Y 1 2 3 4 5 6 7 1 2 3 4 5 6

Univariate lacunary factorization

X Y 1 2 3 4 5 6 7 1 2 3 4 5 6

Multivariate low-degree

factorization

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Weighted-homogeneous factors

X Y 1 2 3 4 5 6 7 1 2 3 4 5 6

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Weighted-homogeneous factors

X Y 1 2 3 4 5 6 7 1 2 3 4 5 6

Reduction to the univariate case

If f, g are (p, q)-homogeneous, g divides f ⇐ ⇒ g(X1/q, 1) divides f(X1/q, 1)

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Weighted-homogeneous factors

X Y 1 2 3 4 5 6 7 1 2 3 4 5 6

Reduction to the univariate case

If f, g are (p, q)-homogeneous, g divides f ⇐ ⇒ g(X1/q, 1) divides f(X1/q, 1) For all possible pairs (p, q):

1. Write f = f1 + · · · + fs as a sum of (p, q)-hom. polynomials;
2. Compute the common degree-d factors of the ft(X1/q, 1)’s;

univariate lacunary factorization (number fields)

3. Return Yp deg(g)g(Xq/Yp) for each factor g.

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The multivariate case

◮ Weighted-homogeneous factors Unidimensional factors:

∃˜ g ∈ K[Z] s.t. g(X1, . . . , Xn) = Xγg(Xδ)

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The multivariate case

◮ Weighted-homogeneous factors Unidimensional factors:

∃˜ g ∈ K[Z] s.t. g(X1, . . . , Xn) = Xγg(Xδ)

For all pairs of monomials (Xα1, Xα2):

1. Write f = f1 + · · · + fs as a sum of unidimensional polynomials;
2. Compute the degree-d factors of the ˜

ft’s; univariate lacunary factorization

3. Return Xγg(Xδ) for each factor g.

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Linear factors of bivariate polynomials

[Chattopadhyay-G.-Koiran-Portier-Strozecki’13]

Observation (Y − uX − v) divides f(X, Y) ⇐ ⇒ f(X, uX + v) ≡ 0

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Linear factors of bivariate polynomials

[Chattopadhyay-G.-Koiran-Portier-Strozecki’13]

Observation (Y − uX − v) divides f(X, Y) ⇐ ⇒ f(X, uX + v) ≡ 0 Theorem val  

ℓ

j=1

cjXαj(uX + v)βj   α1 + ℓ 2

if nonzero and uv = 0.

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Linear factors of bivariate polynomials

[Chattopadhyay-G.-Koiran-Portier-Strozecki’13]

Observation (Y − uX − v) divides f(X, Y) ⇐ ⇒ f(X, uX + v) ≡ 0 Theorem val  

ℓ

j=1

cjXαj(uX + v)βj   α1 + ℓ 2

if nonzero and uv = 0.

Gap Theorem Suppose that f = f1+f2 with valX(f2) > valX(f1)+ ℓ(f1)

2

. Then

for all uv = 0, (Y − uX − v) divides f iff it divides both f1 and f2.

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Puiseux series

Observation for low-degree factors g(X, Y) divides f(X, Y) ⇐ ⇒ f(X, φ(X)) ≡ 0

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Puiseux series

Observation for low-degree factors g(X, Y) divides f(X, Y) ⇐ ⇒ f(X, φ(X)) ≡ 0 g(X, Y) = g0(X)

degY(g)

i=1

(Y − φi(X)) ∈ K(X)[Y]

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Puiseux series

Observation for low-degree factors g(X, Y) divides f(X, Y) ⇐ ⇒ f(X, φ(X)) ≡ 0 g(X, Y) = g0(X)

degY(g)

i=1

(Y − φi(X)) ∈ K(X)[Y]

◮ g0 ∈ K[X] ◮ φ1, . . . , φd ∈ K(X) ⊂ K

X are Puiseux series:

φ(X) =

tt0

atXt/n with at ∈ K, at0 = 0. (val(φ) = t0/n)

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Puiseux series

Observation for low-degree factors g(X, Y) divides f(X, Y) ⇐ ⇒ f(X, φ(X)) ≡ 0 g(X, Y) = g0(X)

degY(g)

i=1

(Y − φi(X)) ∈ K(X)[Y]

◮ g0 ∈ K[X] ◮ φ1, . . . , φd ∈ K(X) ⊂ K

X are Puiseux series:

φ(X) =

tt0

atXt/n with at ∈ K, at0 = 0. (val(φ) = t0/n)

◮ If g is irreducible,

g divides f ⇐ ⇒ ∃i, f(X, φi) = 0 ⇐ ⇒ ∀i, f(X, φi) = 0

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Valuation bound

Theorem Let φ ∈ K X

f valuation v, g of degree d s.t. g(X, φ(X)) = 0,

and f1 = ℓ

j=1 cjXαjYβj.

If the family (Xαjφβj)j is linearly independent, val(f1(X, φ)) min

j (αj + vβj) + (8d2 − v)

ℓ 2

.

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Valuation bound

Theorem Let φ ∈ K X

f valuation v, g of degree d s.t. g(X, φ(X)) = 0,

and f1 = ℓ

j=1 cjXαjYβj.

If the family (Xαjφβj)j is linearly independent, val(f1(X, φ)) min

j (αj + vβj) + (8d2 − v)

ℓ 2

.

◮ Proof based on the Wronskian of the family (Xαjφβj)j. ◮ Optimality?

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Gap Theorem

Gap Theorem Let f =

ℓ

j=1

cjXαjYβj

f1

+

k

j=ℓ+1

cjXαjYβj

f2

with α1 + vβ1 · · · αk + vβk. Let g a degree-d irreducible polynomial, with a root of valuation v. If ℓ is the smallest index s.t. αℓ+1 + vβℓ+1 > (α1 + vβ1) + (8d2 − v) ℓ 2

,

then g divides f iff it divides both f1 and f2.

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Gap Theorem

Gap Theorem Let f =

ℓ

j=1

cjXαjYβj

f1

+

k

j=ℓ+1

cjXαjYβj

f2

with α1 + vβ1 · · · αk + vβk. Let g a degree-d irreducible polynomial, with a root of valuation v. If ℓ is the smallest index s.t. αℓ+1 + vβℓ+1 > (α1 + vβ1) + (8d2 − v) ℓ 2

,

then g divides f iff it divides both f1 and f2.

◮ Depends (only) on v. ◮ Bounds the growth of αj + vβj in f1 (neither αj nor βj)

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Combining two valuations

Gap Theorem for inhomogeneous factors Let f =

ℓ

j=1

cjXαjYβj

f1

+

k

j=ℓ+1

cjXαjYβj

f2

where ℓ is the largest index s.t. for 1 i, j ℓ, |αi − αj|, |βi − βj| (4d4 + 2d2) ℓ − 1 2

.

Then every degree-d inhomogeneous g ∈ K[X, Y], multg(f) = min(multg(f1), multg(f2)).

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An example with d = 1

f = X31Y6 − 2 X30Y7 + X29Y8 − X29Y6 + X18Y13 − X16Y15 + X17Y13 + X16Y14 + X10Y2 − X9Y3 + X9Y2 − X5Y6 + X3Y8 − 2 X3Y7 + X3Y6

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An example with d = 1

f = X31Y6 − 2 X30Y7 + X29Y8 − X29Y6 + X18Y13 − X16Y15 + X17Y13 + X16Y14 + X10Y2 − X9Y3 + X9Y2 − X5Y6 + X3Y8 − 2 X3Y7 + X3Y6 X Y 5 10 15 20 25 30 5 10 15

•
•
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Bruno Grenet – Computing low-degree factors of lacunary polynomials

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An example with d = 1

f = X31Y6 − 2 X30Y7 + X29Y8 − X29Y6 + X18Y13 − X16Y15 + X17Y13 + X16Y14 + X10Y2 − X9Y3 + X9Y2 − X5Y6 + X3Y8 − 2 X3Y7 + X3Y6 X Y 5 10 15 20 25 30 5 10 15

•
•
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Bruno Grenet – Computing low-degree factors of lacunary polynomials

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An example with d = 1

f = X31Y6 − 2 X30Y7 + X29Y8 − X29Y6 + X18Y13 − X16Y15 + X17Y13 + X16Y14 + X10Y2 − X9Y3 + X9Y2 − X5Y6 + X3Y8 − 2 X3Y7 + X3Y6 X Y 5 10 15 20 25 30 5 10 15

•
•
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Bruno Grenet – Computing low-degree factors of lacunary polynomials

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An example with d = 1

f = X31Y6 − 2 X30Y7 + X29Y8 − X29Y6 + X18Y13 − X16Y15 + X17Y13 + X16Y14 + X10Y2 − X9Y3 + X9Y2 − X5Y6 + X3Y8 − 2 X3Y7 + X3Y6 X Y 5 10 15 20 25 30 5 10 15

•
•
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An example with d = 1

f = X31Y6 − 2 X30Y7 + X29Y8 − X29Y6 + X18Y13 − X16Y15 + X17Y13 + X16Y14 + X10Y2 − X9Y3 + X9Y2 − X5Y6 + X3Y8 − 2 X3Y7 + X3Y6 f1 = X3Y6(−X2 + Y2 − 2Y + 1)

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An example with d = 1

f = X31Y6 − 2 X30Y7 + X29Y8 − X29Y6 + X18Y13 − X16Y15 + X17Y13 + X16Y14 + X10Y2 − X9Y3 + X9Y2 − X5Y6 + X3Y8 − 2 X3Y7 + X3Y6 f1 = X3Y6(X − Y + 1)(1 − X − Y)

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An example with d = 1

f = X31Y6 − 2 X30Y7 + X29Y8 − X29Y6 + X18Y13 − X16Y15 + X17Y13 + X16Y14 + X10Y2 − X9Y3 + X9Y2 − X5Y6 + X3Y8 − 2 X3Y7 + X3Y6 f1 = X3Y6(X − Y + 1)(1 − X − Y) f2 = X9Y2(X − Y + 1) f3 = X16Y13(X + Y)(X − Y + 1) f4 = X29Y6(X + Y − 1)(X − Y + 1)

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An example with d = 1

f = X31Y6 − 2 X30Y7 + X29Y8 − X29Y6 + X18Y13 − X16Y15 + X17Y13 + X16Y14 + X10Y2 − X9Y3 + X9Y2 − X5Y6 + X3Y8 − 2 X3Y7 + X3Y6 f1 = X3Y6(X − Y + 1)(1 − X − Y) f2 = X9Y2(X − Y + 1) f3 = X16Y13(X + Y)(X − Y + 1) f4 = X29Y6(X + Y − 1)(X − Y + 1) = ⇒ linear factors of f: (X − Y + 1, 1)

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An example with d = 1

f = X31Y6 − 2 X30Y7 + X29Y8 − X29Y6 + X18Y13 − X16Y15 + X17Y13 + X16Y14 + X10Y2 − X9Y3 + X9Y2 − X5Y6 + X3Y8 − 2 X3Y7 + X3Y6 f1 = X3Y6(X − Y + 1)(1 − X − Y) f2 = X9Y2(X − Y + 1) f3 = X16Y13(X + Y)(X − Y + 1) f4 = X29Y6(X + Y − 1)(X − Y + 1) = ⇒ linear factors of f: (X − Y + 1, 1), (X, 3), (Y, 2)

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Algorithm

Theorem Given f ∈ K[X, Y] in lacunary representation, one can compute in time poly(size(f), d) a degree-O(d4k2) polynomial fld s.t. for all inhomogeneous degree-d polynomial g, multg(f) = multg(fld).

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Algorithm

Theorem Given f ∈ K[X1, . . . , Xn] in lacunary representation, one can com- pute in time poly(size(f), d) a degree-O(d4k2) polynomial fld s.t. for all inhomogeneous degree-d polynomial g, multg(f) = multg(fld).

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Algorithm

Theorem Given f ∈ K[X1, . . . , Xn] in lacunary representation, one can com- pute in time poly(size(f), d) a degree-O(d4k2) polynomial fld s.t. for all inhomogeneous degree-d polynomial g, multg(f) = multg(fld).

1. Write f = f1 + · · · + fs where

degXi(ft) − valXi(ft) (4d4 − 2d2) ℓt

2

for all i;
2. Return gcd(f1, . . . , ft).
3. (Factor the gcd using a low-degree factorization algorithm.)

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Complete algorithm

Find degree-d factors of f =

k

j=1

cjXαj (Xi, minj αi,j) Degree-d factors

f univariate

lacunary polynomials Factors of fld

f degree O(d4k2)

monomials unidim. multidim.

Available for Q(α) only Impossible for Q, C Low-degree factorization Q(α), Q, R, C, Qp, etc.

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Complete algorithm

Find degree-d factors of f =

k

j=1

cjXαj (Xi, minj αi,j) Degree-d factors

f univariate

lacunary polynomials Factors of fld

f degree O(d4k2)

monomials unidim. multidim.

Available for Q(α) only Impossible for Q, C Low-degree factorization Q(α), Q, R, C, Qp, etc.

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Complete algorithm

Find degree-d factors of f =

k

j=1

cjXαj (Xi, minj αi,j) Degree-d factors

f univariate

lacunary polynomials Factors of fld

f degree O(d4k2)

monomials unidim. multidim.

Available for Q(α) only Impossible for Q, C Low-degree factorization Q(α), Q, R, C, Qp, etc.

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Complete algorithm

Find degree-d factors of f =

k

j=1

cjXαj (Xi, minj αi,j) Degree-d factors

f univariate

lacunary polynomials Factors of fld

f degree O(d4k2)

monomials unidim. multidim.

Available for Q(α) only Impossible for Q, C Low-degree factorization Q(α), Q, R, C, Qp, etc.

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Implementation in progress

http://www.mathemagix.org/ > Packages > Lacunaryx

Factorization-related algorithms for lacunary polynomials

◮ Integer roots of lacunary univariate polynomials ◮ Linear factors of lacunary univariate and bivariate polynomials ◮ Bounded-degree factors: in progress ◮ Very large degree polynomials (G. Lecerf)

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Conclusion

◮ Computing low-degree factors of lacunary multivariate polynomials

Reduction to
univariate lacunary polynomials

low-degree multivariate polynomials

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Conclusion

◮ Computing low-degree factors of lacunary multivariate polynomials

Reduction to
univariate lacunary polynomials

low-degree multivariate polynomials

“Field-independent”
Simpler and more general than previous algorithms
Partial results in large positive characteristic
Implementation: work in progress

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Conclusion

◮ Computing low-degree factors of lacunary multivariate polynomials

Reduction to
univariate lacunary polynomials

low-degree multivariate polynomials

“Field-independent”
Simpler and more general than previous algorithms
Partial results in large positive characteristic
Implementation: work in progress

◮ Open questions:

Lacunary factors in polynomial time?
More general settings: SLP/arithmetic circuits

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Conclusion

◮ Computing low-degree factors of lacunary multivariate polynomials

Reduction to
univariate lacunary polynomials

low-degree multivariate polynomials

“Field-independent”
Simpler and more general than previous algorithms
Partial results in large positive characteristic
Implementation: work in progress

◮ Open questions:

Lacunary factors in polynomial time?
More general settings: SLP/arithmetic circuits
Degree-d factors in positive characteristic?
Small positive characteristic?

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Conclusion

◮ Computing low-degree factors of lacunary multivariate polynomials

Reduction to
univariate lacunary polynomials

low-degree multivariate polynomials

“Field-independent”
Simpler and more general than previous algorithms
Partial results in large positive characteristic
Implementation: work in progress

◮ Open questions:

Lacunary factors in polynomial time?
More general settings: SLP/arithmetic circuits
Degree-d factors in positive characteristic?
Small positive characteristic?

Thank you!

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