coprime polynomial pairs, hankel matrices, and splitting...

Coprime polynomial pairs, Hankel matrices, andsplitting subspaces

Sudhir R. Ghorpade

Department of MathematicsIndian Institute of Technology Bombay

Powai, Mumbai 400076, Indiahttp://www.math.iitb.ac.in/∼srg/

CanaDAM-2011Victoria, BC, Canada

June 1, 2011

Sudhir R. Ghorpade Coprime polynomial pairs, Hankel matrices, and splitting subspaces

References and Names of collaborators

This talk corresponds mainly to the following two papers:

Mario Garcia-Armas, Sudhir R. Ghorpade, and Samrith Ram.Relatively prime polynomials and nonsingular Hankel matricesover finite fields, Journal of Combinatorial Theory, Series A,Vol. 118, No. 3 (2011), pp. 819-828.

Sudhir R. Ghorpade, and Samrith Ram. Block companionSinger cycles, primitive recursive vector sequences, andcoprime polynomial pairs over finite fields, Finite Fields andTheir Applications, (2011), doi:10.1016/j.ffa.2011.02.008.

A precursor to the latter was the following paper:

Sudhir R. Ghorpade, Sartaj Ul Hasan and Meena Kumari,Primitive polynomials, Singer cycles, and word-oriented linearfeedback shift registers, Designs, Codes and Cryptography,Vol. 58, No. 2 (2011), pp. 123-134.

All these (and more!) are available at:http://www.math.iitb.ac.in/∼srg/Papers.html

Sudhir R. Ghorpade Coprime polynomial pairs, Hankel matrices, and splitting subspaces

Some Questions

What is the probability that two polynomials in Fq[X ] ofdegree n ≥ 1 are relatively prime?

What is the probability that a n × n Hankel matrix over Fq isnonsingular? Similarly, what is the probability that a n × nToeplitz matrix over Fq is nonsingular?[Recall: A = (aij) is Hankel if aij = ars whenever i + j = r + s,and it is Toeplitz if aij = ars whenever i − j = r − s.]

(Niederreiter, 1995) Given σ such that Fqmn = Fq(σ), what isthe number of σ-splitting subspaces of Fqmn of dimension m?[Recall: An m-dimensional subspace W of Fqmn is σ-splitting if

Fqmn = W ⊕ σW ⊕ · · · ⊕ σn−1W . ]

Sudhir R. Ghorpade Coprime polynomial pairs, Hankel matrices, and splitting subspaces

Some Questions

What is the probability that two polynomials in Fq[X ] ofdegree n ≥ 1 are relatively prime?

What is the probability that a n × n Hankel matrix over Fq isnonsingular? Similarly, what is the probability that a n × nToeplitz matrix over Fq is nonsingular?[Recall: A = (aij) is Hankel if aij = ars whenever i + j = r + s,and it is Toeplitz if aij = ars whenever i − j = r − s.]

(Niederreiter, 1995) Given σ such that Fqmn = Fq(σ), what isthe number of σ-splitting subspaces of Fqmn of dimension m?[Recall: An m-dimensional subspace W of Fqmn is σ-splitting if

Fqmn = W ⊕ σW ⊕ · · · ⊕ σn−1W . ]

Sudhir R. Ghorpade Coprime polynomial pairs, Hankel matrices, and splitting subspaces

Some Questions

What is the probability that two polynomials in Fq[X ] ofdegree n ≥ 1 are relatively prime?

What is the probability that a n × n Hankel matrix over Fq isnonsingular? Similarly, what is the probability that a n × nToeplitz matrix over Fq is nonsingular?[Recall: A = (aij) is Hankel if aij = ars whenever i + j = r + s,and it is Toeplitz if aij = ars whenever i − j = r − s.]

(Niederreiter, 1995) Given σ such that Fqmn = Fq(σ), what isthe number of σ-splitting subspaces of Fqmn of dimension m?[Recall: An m-dimensional subspace W of Fqmn is σ-splitting if

Fqmn = W ⊕ σW ⊕ · · · ⊕ σn−1W . ]

Sudhir R. Ghorpade Coprime polynomial pairs, Hankel matrices, and splitting subspaces

Coprime Polynomial Pairs

Thanks to Knuth [ACP-2 (§4.6.1, Ex. 5), 1969] and more recently,Corteel, Savage, Wilf, Zeilberger (and Zagier) [JCT-A, 1998], weknow that the probability that two monic polynomials of degreen ≥ 1 over Fq, chosen independently and uniformly at random, arerelatively prime is

1− 1

q.

A bijective “explanation” was given by Reifegerste (2000) in thecase q = 2. The case of arbitrary q was “explained” by Benjaminand Bennett (2007) by constructing an explicit surjective map

{(f , g) : f , g ∈ Fq[X ] monic, deg n, coprime}→ {(f , g) : f , g monic, deg n, non-coprime}

such that the cardinality of each fiber is q − 1.

Sudhir R. Ghorpade Coprime polynomial pairs, Hankel matrices, and splitting subspaces

Coprime Polynomial Pairs

Thanks to Knuth [ACP-2 (§4.6.1, Ex. 5), 1969] and more recently,Corteel, Savage, Wilf, Zeilberger (and Zagier) [JCT-A, 1998], weknow that the probability that two monic polynomials of degreen ≥ 1 over Fq, chosen independently and uniformly at random, arerelatively prime is

1− 1

q.

A bijective “explanation” was given by Reifegerste (2000) in thecase q = 2. The case of arbitrary q was “explained” by Benjaminand Bennett (2007) by constructing an explicit surjective map

{(f , g) : f , g ∈ Fq[X ] monic, deg n, coprime}→ {(f , g) : f , g monic, deg n, non-coprime}

such that the cardinality of each fiber is q − 1.

Sudhir R. Ghorpade Coprime polynomial pairs, Hankel matrices, and splitting subspaces

Side Remark: Connection with Riemann Zeta Function

It is an elementary and well-known that the probability of twointegers to be relatively prime is

1

ζ(2)=

6

π2.

To note that the result for polynomials fits in with this, recall thatthe Weil and Riemann zeta functions of any variety V over Fq are

ZV (T ) := exp

( ∞∑n=1

|V (Fqn)| T n

n

)and ζV (s) := ZV

(q−s).

Now Fq[X ]←→ A1Fq

and

ZA1(T ) = exp

( ∞∑n=1

qn T n

n

)= exp

(log(1− qT )−1

)=

1

1− qT

=⇒ ζA1(2) =1

1− q(q−2)and

1

ζA1(2)= 1− 1

q.

Sudhir R. Ghorpade Coprime polynomial pairs, Hankel matrices, and splitting subspaces

Side Remark: Connection with Riemann Zeta Function

It is an elementary and well-known that the probability of twointegers to be relatively prime is

1

ζ(2)=

6

π2.

To note that the result for polynomials fits in with this, recall thatthe Weil and Riemann zeta functions of any variety V over Fq are

ZV (T ) := exp

( ∞∑n=1

|V (Fqn)| T n

n

)and ζV (s) := ZV

(q−s).

Now Fq[X ]←→ A1Fq

and

ZA1(T ) = exp

( ∞∑n=1

qn T n

n

)= exp

(log(1− qT )−1

)=

1

1− qT

=⇒ ζA1(2) =1

1− q(q−2)and

1

ζA1(2)= 1− 1

q.

Sudhir R. Ghorpade Coprime polynomial pairs, Hankel matrices, and splitting subspaces

Side Remark: Connection with Riemann Zeta Function

It is an elementary and well-known that the probability of twointegers to be relatively prime is

1

ζ(2)=

6

π2.

To note that the result for polynomials fits in with this, recall thatthe Weil and Riemann zeta functions of any variety V over Fq are

ZV (T ) := exp

( ∞∑n=1

|V (Fqn)| T n

n

)and ζV (s) := ZV

(q−s).

Now Fq[X ]←→ A1Fq

and

ZA1(T ) = exp

( ∞∑n=1

qn T n

n

)= exp

(log(1− qT )−1

)=

1

1− qT

=⇒ ζA1(2) =1

1− q(q−2)and

1

ζA1(2)= 1− 1

q.

Sudhir R. Ghorpade Coprime polynomial pairs, Hankel matrices, and splitting subspaces

Nonsingular Hankel Matrices

Daykin [Crelle, 1960] was perhaps the first to prove that thenumber of n × n nonsingular Hankel matrices over Fq isq2n−2(q − 1). In particular, the probability that a Hankel matrixover Fq is nonsingular is

1− 1

q.

[More generally, Daykin has formulas for the number of m × nHankel matrices over Fq of a given rank r ≤ min{m, n}. ]

Recently, Daykin’s formula has been (re)proved [usingsubresultants and such] by Kaltofen and Lobo (1996) in thecontext of counting n× n nonsingular Toeplitz matrices over Fq. Asimpler and more direct proof can be found in our paper in JCT-A118 (2011), 819828. The case of Hankel matrices over Fq of agiven rank is also discussed there.

Sudhir R. Ghorpade Coprime polynomial pairs, Hankel matrices, and splitting subspaces

Coprime Polynomial Pairs and Nonsingular Hankel Matrices

Curious Fact: The probability for a polynomial pair to be coprimeand for a Hankel matrix to be nonsingular is the same! In fact,

|CPPn(Fq)| = q2n

(1− 1

q

)= q2n−1(q − 1),

and

|HGLn(Fq)| = q2n−1(

1− 1

q

)= q2n−2(q − 1),

where CPPn(Fq) denotes the set of all ordered pairs of coprimemonic polynomials over Fq of degree n and HGLn(Fq) denotes theset of all n × n nonsingular Hankel matrices with entries in Fq.

Question: Can we “explain” this fact? In other words, can we givea combinatorial proof that |CPPn(Fq)| = q |HGLn(Fq)|?

Sudhir R. Ghorpade Coprime polynomial pairs, Hankel matrices, and splitting subspaces

Coprime Polynomial Pairs and Nonsingular Hankel Matrices

Curious Fact: The probability for a polynomial pair to be coprimeand for a Hankel matrix to be nonsingular is the same! In fact,

|CPPn(Fq)| = q2n

(1− 1

q

)= q2n−1(q − 1),

and

|HGLn(Fq)| = q2n−1(

1− 1

q

)= q2n−2(q − 1),

where CPPn(Fq) denotes the set of all ordered pairs of coprimemonic polynomials over Fq of degree n and HGLn(Fq) denotes theset of all n × n nonsingular Hankel matrices with entries in Fq.

Question: Can we “explain” this fact? In other words, can we givea combinatorial proof that |CPPn(Fq)| = q |HGLn(Fq)|?

Sudhir R. Ghorpade Coprime polynomial pairs, Hankel matrices, and splitting subspaces

An explicit surjection

Let F be any field and n be any positive integer.

Theorem

There is a surjective map σ : CPPn(F )→ HGLn(F ) such thatfor any A ∈ HGLn(F ), the fiber σ−1 ({A}) is in one-to-onecorrespondence with F . In particular, |CPPn(Fq)| = q |HGLn(Fq)| .

Sketch of Proof: A key idea is to consider the Bezoutian. Recallthat the Bezoutian (matrix) of u, v ∈ F [X ] of degree ≤ n is then × n matrix Bn(u, v) = (bij) determined by the equation

u(X )v(Y )− v(X )u(Y )

X − Y=

n∑i ,j=1

bijXi−1Y j−1.

Fact: Assume that deg u = n and deg v ≤ n. Then

Bn(u, v) is nonsingular ⇐⇒ u and v are coprime

Sudhir R. Ghorpade Coprime polynomial pairs, Hankel matrices, and splitting subspaces

An explicit surjection

Let F be any field and n be any positive integer.

Theorem

There is a surjective map σ : CPPn(F )→ HGLn(F ) such thatfor any A ∈ HGLn(F ), the fiber σ−1 ({A}) is in one-to-onecorrespondence with F . In particular, |CPPn(Fq)| = q |HGLn(Fq)| .

Sketch of Proof: A key idea is to consider the Bezoutian. Recallthat the Bezoutian (matrix) of u, v ∈ F [X ] of degree ≤ n is then × n matrix Bn(u, v) = (bij) determined by the equation

u(X )v(Y )− v(X )u(Y )

X − Y=

n∑i ,j=1

bijXi−1Y j−1.

Fact: Assume that deg u = n and deg v ≤ n. Then

Bn(u, v) is nonsingular ⇐⇒ u and v are coprime

Sudhir R. Ghorpade Coprime polynomial pairs, Hankel matrices, and splitting subspaces

Illustration of the Bezoutian

Example: Suppose u(X ) = u0 + u1X + · · ·+ unX n withu0, u1, . . . , un ∈ F and v(X ) = 1. Then

u(X )− u(Y )

X − Y=

n∑k=1

ukX k − Y k

X − Y=

n∑k=1

uk

k∑i=1

X i−1 Y k−i

=n∑

i ,j=1

ui+j−1X i−1 Y j−1,

where, by convention, uk := 0 for k > n. Thus

Bn(u, 1) =

u1 u2 · · · un−1 un

u2 u3 · · · un 0... . . . ...

un−1 un · · · 0 0un 0 · · · 0 0

In particular, if deg u = n, i.e., if un 6= 0, then u and v are coprime,and moreover Bn(u, v) is nonsingular.

Sudhir R. Ghorpade Coprime polynomial pairs, Hankel matrices, and splitting subspaces

Pade pairs and Hermite pairs

Define the set of Pade pairs by

Pn(F ) :={

(u, v) ∈ F [X ]2 : u monic, deg u = n, and deg v < n},

and the set of Hermite pairs by

HPn(F ) := {(u, v) ∈ Pn(F ) : u and v are coprime} .

Observe that CPPn(F ) is in bijection with HPn(F ) [Proof: Themap given by (f , g) 7→ (f , g − f ) does the job.]. Next, observe thatfor any (u, v) ∈ Pn(F ), there are unique ai ∈ F , i ≥ 1, such that

v(X )

u(X )=∞∑i=1

aiX i.

Define Hn(u, v) to be the n × n Hankel matrix (ai+j−1).

Sudhir R. Ghorpade Coprime polynomial pairs, Hankel matrices, and splitting subspaces

Completion of the proof

Lemma (Barnett factorization)

For any (u, v) ∈ Pn(F ),

Bn(u, v) = Bn(u, 1)Hn(u, v)Bn(u, 1).

Thanks to the above results, (u, v) 7−→ Hn(u, v) gives a mapHPn(F )→ HGLn(F ). With some effort, we can show that it issurjective and that the fibre is in bijection with F . Combining thiswith the bijection CPPn(F )→ HPn(F ), we obtain the desiredresult.

Remark: One can replace HGLn(F ) by the set TGLn(F ) of n × nnonsingular Toeplitz matrices over Fq. Indeed if E denotes then × n matrix with 1 on the antidiagonal and 0 elsewhere, then E isnonsingular and the map given by A 7→ AE gives a bijectionbetween TGLn(F ) and HGLn(F ).

Sudhir R. Ghorpade Coprime polynomial pairs, Hankel matrices, and splitting subspaces

Completion of the proof

Lemma (Barnett factorization)

For any (u, v) ∈ Pn(F ),

Bn(u, v) = Bn(u, 1)Hn(u, v)Bn(u, 1).

Thanks to the above results, (u, v) 7−→ Hn(u, v) gives a mapHPn(F )→ HGLn(F ). With some effort, we can show that it issurjective and that the fibre is in bijection with F . Combining thiswith the bijection CPPn(F )→ HPn(F ), we obtain the desiredresult.

Remark: One can replace HGLn(F ) by the set TGLn(F ) of n × nnonsingular Toeplitz matrices over Fq. Indeed if E denotes then × n matrix with 1 on the antidiagonal and 0 elsewhere, then E isnonsingular and the map given by A 7→ AE gives a bijectionbetween TGLn(F ) and HGLn(F ).

Sudhir R. Ghorpade Coprime polynomial pairs, Hankel matrices, and splitting subspaces

Splitting Subspaces

Let m, n be positive integers and σ ∈ Fqmn . Recall that anm-dimensional subspace W of Fqmn is σ-splitting if

Fqmn = W ⊕ σW ⊕ · · · ⊕ σn−1W . ]

Denote by S(σ,m, n; q) the number of m-dimensional σ-splittingsubspaces of Fqmn .

Note: For an arbitrary σ ∈ Fqmn , there may not be any σ-splittingsubspace; for example, this happens if σ ∈ Fq and n > 1. However,if σ ∈ Fqmn satisfies Fqmn = Fq(σ), then a σ-splitting subspaceexists and, in fact, S(σ,m, n; q) ≥ (qmn − 1)/(qm − 1).

Examples: If min{m, n} = 1 and if σ ∈ Fqmn is such thatFqmn = Fq(σ), then it is easy to see that

S(σ,m, n; q) =qmn − 1

qm − 1.

Sudhir R. Ghorpade Coprime polynomial pairs, Hankel matrices, and splitting subspaces

Quantitative Formulation of Niederreiter’s Question

Splitting Subspace Conjecture: For any σ such that Fqmn = Fq(σ),

S(σ,m, n; q) =qmn − 1

qm − 1qm(m−1)(n−1).

One can have a simpler formulation using the following notion:

Definition

By a pointed σ-splitting subspace of dimension m we shall mean apair (W , x) where W is an m-dimensional σ-splitting subspace ofFqmn and x ∈W . The element x may be referred to as the basepoint of (W , x).

Pointed Splitting Subspace Conjecture: For any x ∈ F∗qmn and anyσ such that Fqmn = Fq(σ), the number of m-dimensional pointed

σ-splitting subspaces of Fqmn with base point x is qm(m−1)(n−1).Consequently, there is a one-to-one correspondence betweenpointed splitting subspaces and pointed n-tuples of m ×mnilpotent matrices over Fq.

Sudhir R. Ghorpade Coprime polynomial pairs, Hankel matrices, and splitting subspaces

What do we know about the SSC and PSSC?

The case min{m, n} = 1 is easy. The best known general resultseems to be the following.

Theorem

The Splitting Subspace Conjecture as well as the Pointed SplittingSubspace Conjecture holds in the affirmative if m = 2.

We remark that one of the key ingredients in the proof of thistheorem is a result closely related to the question about theprobability for a pair of polynomials in Fq[X ] to be relatively prime.More precisely, we use a result of Benjamin and Bennett about thenumber of coprime pairs of polynomials in Fq[X ] of degree < n.

Sudhir R. Ghorpade Coprime polynomial pairs, Hankel matrices, and splitting subspaces

Bounds and Asymptotics

For any σ ∈ Fqmn , defineN(σ,m, n; q) := S(σ,m, n; q)

∏m−1i=0 (qm − qi ).

Theorem

Let σ ∈ Fqmn be such that Fqmn = Fq(σ).

(q − 2)qmn + 1

(q − 1)qmn(m−1) ≤ N(σ,m, n; q) ≤

m−1∏i=0

(qmn − qi ).

Corollary

Let σ ∈ Fqmn be such that Fqmn = Fq(σ) and let x ∈ F∗qmn . Thenthe number of m-dimensional pointed σ-splitting subspaces of Fqmn

with base point x is is asymptotically equivalent to qm(m−1)(n−1)

as q →∞.

Sudhir R. Ghorpade Coprime polynomial pairs, Hankel matrices, and splitting subspaces

Related Problems

The above conjecture is closely related to:

1 Counting the number of primitive σ-LFSRs of order n overFqm , or equivalently, the number of primitive recursive vectorsequences of order n over Fm

q .2 Counting the number of (m, n)-block companion Singer cycles

over Fq, i.e., the number of mn ×mn block matrices T of thefollowing form which are nonsingular and whose order inGLmn(Fq) is the meximum possible:

T =

0 0 0 . . 0 0 C0

Im 0 0 . . 0 0 C1

. . . . . . . .

. . . . . . . .0 0 0 . . Im 0 Cn−20 0 0 . . 0 Im Cn−1

,

where C0,C1, . . . ,Cn−1 ∈ Mm(Fq) and Im and 0 denote them ×m identity and zero matrix over Fq, respectively.

Sudhir R. Ghorpade Coprime polynomial pairs, Hankel matrices, and splitting subspaces

Thank you for your attention!

Sudhir R. Ghorpade Coprime polynomial pairs, Hankel matrices, and splitting subspaces