Schinzel's Theorem on Factoring Sparse Polynomials - Prof. Ma Filaseta, Study notes of Mathematics

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FACTORING S PARSE P OLYNOMIALS

Theorem 1 (Schinzel): Let r be a positive integer, and fix

non-zero integers a 0

,... , a r

. Let

F (x 1

,... , x r

) = a r

x r

• · · · + a 1

x 1

• a 0

Then there exist finite sets S and T of matrices satisfying:

Theorem 1 (Schinzel): Let r be a positive integer, and fix

non-zero integers a 0

,... , a r

. Let

F (x 1

,... , x r

) = a r

x r

• · · · + a 1

x 1

• a 0

Then there exist finite sets S and T of matrices satisfying:

(i) Each matrix in S or T is an r × ρ matrix with

integer entries and of rank ρ for some ρ ≤ r.

(ii) The matrices in S and T are computable.

(iii) For every set of positive integers d 1

,... , d r

with d 1

< d 2

< · · · < d r

, the non-reciprocal

part of F (x

d 1 ,... , x

d r ) is reducible if and

only if there is an r × ρ matrix N in S and

integers v 1

,... , v ρ

satisfying

d 1

d 2

d r

= N

v 1

v 2

v ρ

but there is no r × ρ

′

matrix M in T with

ρ

′

< ρ and no integers v

′

1

,... , v

′

ρ

′

satisfying

d 1

d 2

d r

= M

v

′

1

v

′

2

v

′

ρ

′

the non-reciprocal part of F (x

d 1 ,... , x

d r ) is reducible

F (x 1

,... , x r

) = a r

x r

• · · · + a 1

x 1

• a 0

the non-reciprocal part of F (x

d 1 ,... , x

d r ) is reducible

F (x 1

,... , x r

) = a r

x r

• · · · + a 1

x 1

• a 0

F (x

d 1 ,... , x

d r

) = a r

x

d r

• · · · + a 1

x

d 1

• a 0

Theorem 2 (Schinzel): Let r be a positive integer, and fix

non-zero integers a 0

,... , a r

. Let

F (x 1

,... , x r

) = a r

x r

• · · · + a 1

x 1

• a 0

Then there exist finite sets S and T of matrices satisfying:

(i) Each matrix in S or T is an r × ρ matrix with

integer entries and of rank ρ for some ρ ≤ r.

Theorem 2 (Schinzel): Let r be a positive integer, and fix

non-zero integers a 0

,... , a r

. Let

F (x 1

,... , x r

) = a r

x r

• · · · + a 1

x 1

• a 0

Then there exist finite sets S and T of matrices satisfying:

(i) Each matrix in S or T is an r × ρ matrix with

integer entries and of rank ρ for some ρ ≤ r.

(ii) The matrices in S and T are computable.

(iii) For every set of positive integers d 1

,... , d r

with F (x

d 1 ,... , x

d r ) not reciprocal and

d 1

< d 2

< · · · < d r

, the non-cyclotomic

part of F (x

d 1 ,... , x

d r ) is reducible if and

only if there is an r × ρ matrix N in S and

integers v 1

,... , v ρ

satisfying

d 1

d 2

d r

= N

v 1

v 2

v ρ

but there is no r × ρ

′

matrix M in T with

ρ

′

< ρ and no integers v

′

1

,... , v

′

ρ

′

satisfying

d 1

d 2

d r

= M

v

′

1

v

′

2

. . .

v

′

ρ

′

(iii) For every set of positive integers d 1

,... , d r

with F (x

d 1 ,... , x

d r ) not reciprocal and

d 1

< d 2

< · · · < d r

, the non-cyclotomic

part of F (x

d 1 ,... , x

d r ) is reducible if and

only if there is an r × ρ matrix N in S and

integers v 1

,... , v ρ

satisfying

d 1

d 2

d r

= N

v 1

v 2

v ρ

but there is no r × ρ

′

matrix M in T with

ρ

′

< ρ and no integers v

′

1

,... , v

′

ρ

′

satisfying

d 1

d 2

d r

= M

v

′

1

v

′

2

. . .

v

′

ρ

′

Theorem: There is an algorithm with the following prop-

erty: Given a non-reciprocal f (x) ∈ Z[x] with N non-

zero terms, degree n and height H, the algorithm deter-

mines whether f (x) is irreducible in time

c(N, H)(log n)

c

′

(N )

where c(N, H) depends only on N and H and c

′

(N )

depends only on N.

Proof. Let

f (x) = a r

x

d r

• · · · + a 1

x

d 1

• a 0

Proof. Let

f (x) = a r

x

d r

• · · · + a 1

x

d 1

• a 0

Consider

F (x 1

,... , x r

) = a r

x r

• · · · + a 1

x 1

• a 0

so that

F (x

d 1 ,... , x

d r

) = a r

x

d r

• · · · + a 1

x

d 1

• a 0

Begin the algorithm by constructing the finite sets S and T

of matrices in Schinzel’s Theorem 2.

Proof. Let

f (x) = a r

x

d r

• · · · + a 1

x

d 1

• a 0

Consider

F (x 1

,... , x r

) = a r

x r

• · · · + a 1

x 1

• a 0

so that

F (x

d 1 ,... , x

d r

) = a r

x

d r

• · · · + a 1

x

d 1

• a 0

Begin the algorithm by constructing the finite sets S and T

of matrices in Schinzel’s Theorem 2. Observe that S and

T depend on F and not on the d 1

,... , d r