Polynomial Functions of the Ring of Dual Numbers Modulo · Background Dual Numbers Null polynomials...

BackgroundDual Numbers

Null polynomials over Zm [α]Polynomial Functions over Zm [α]

Counting FormulasSome Generalizations

References

Polynomial Functions of the Ring of DualNumbers Modulo m

Amr Al-Maktry1 Hasan Al-Ezeh2 Sophie Frisch1

1TU Graz University

2Jordan University

Conference on Rings and Factorizations, 2018

Amr Al-Maktry, Hasan Al-Ezeh, Sophie Frisch Polynomial Functions of the Ring of Dual Numbers Modulo m

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Outline

1 Background

2 Dual Numbers

3 Null polynomials over Zm[α]

4 Polynomial Functions over Zm[α]

5 Counting Formulas

6 Some Generalizations

References

Background

Let R be a finite commutative ring with unity.

A function F : R −→ R is said to be a polynomial functionover R if there exists a polynomial f (x) ∈ R[x ] such thatf (a) = F (a) for every a ∈ R. In this case we say that F is theinduced function of f (x) over R and f (x) represents (induces)F .

If F is a bijection then F is called a permutation polynomial.

Let f (x) ∈ R[x ] such that f (a) = 0 for every a ∈ R. f (x) iscalled null polynomial over R. In particular if R = Zm, f (x)called null polynomial (mod m).

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Background

Let f (x) ∈ R[x ] such that f (a) = 0 for every a ∈ R. f (x) iscalled null polynomial over R.

In particular if R = Zm, f (x)called null polynomial (mod m).

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Background

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Background

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Background

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Background

Let f (x) ∈ R[x ] such that f (a) = 0 for every a ∈ R. f (x) iscalled null polynomial over R. In particular if R = Zm, f (x)called null polynomial (mod m).

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Background

Throughout:

F(R) denote the set of polynomial functions over R.

P(R) denote the set of permutation polynomials over R.

µ(m) denote the Kempner’s function, the smallest positiveinteger such that m divides µ(m)!.

f ′(x) denote the formal derivative of f (x).

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Background

Throughout:

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Background

Throughout:

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Background

Throughout:

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Background

Throughout:

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Dual Numbers

When R is a commutative ring, then R[α] designates the result of

adjoint α to R with α2 = 0; that is, R[α] is R[x ]/(x2), where α

denote x + (x2).

Simply R[α] = {a + bα : a, b ∈ R}

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Dual Numbers

When R is a commutative ring, then R[α] designates the result of

adjoint α to R with α2 = 0; that is, R[α] is R[x ]/(x2), where α

denote x + (x2). Simply R[α] = {a + bα : a, b ∈ R}

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Let R be a commutative ring, then1 For a, a′, b, b′ ∈ R. We have

(a + bα)(a′ + b′α) = aa′ + (ab′ + a′b)α(a + bα) is a unit in R[α] iff a is a unit in R.f (a + bα) = f (a) + bf ′(a)α for every f (x) ∈ R[x ]

2 R[α] is a local ring iff R is a local ring.

3 If R is a local ring with a maximal ideal m has nilpotency n.then R[α] is a local ring whose maximal ideal m + αR hasnilpotency n + 1

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(a + bα)(a′ + b′α) = aa′ + (ab′ + a′b)α

(a + bα) is a unit in R[α] iff a is a unit in R.f (a + bα) = f (a) + bf ′(a)α for every f (x) ∈ R[x ]

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(a + bα)(a′ + b′α) = aa′ + (ab′ + a′b)α(a + bα) is a unit in R[α] iff a is a unit in R.

f (a + bα) = f (a) + bf ′(a)α for every f (x) ∈ R[x ]

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Dual Numbers

Definition (Frisch (1999))

Let R be a finite commutative local ring with a maximal ideal mwhose nilpotency K ∈ N. We call R suitable, if for all a, b ∈ R andall l ∈ N, ab ∈ ml ⇒ a ∈ mi and b ∈ mj with i + j ≥ min(K , l).

Proposition

Let R be a finite commutative local ring. Then R[α] is suitable iffR is a field. In particular Zpn [α] is suitable iff n = 1.

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Dual Numbers

Proposition

Let R be a finite commutative local ring. Then R[α] is suitable iffR is a field.

In particular Zpn [α] is suitable iff n = 1.

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Dual Numbers

Proposition

Let R be a finite commutative local ring. Then R[α] is suitable iffR is a field. In particular Zpn [α] is suitable iff n = 1.

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Null polynomials over Zm[α]

Proposition

Suppose that f (x) = f1(x) + f2(x)α, where f1(x), f2(x) ∈ Z[x ].Then f (x) is a null polynomial over Zm[α] iff f1(x), f ′1(x) andf2(x) are null polynomials modulo m.

Corollary

f (x) = (x)2µ(m) =∏2µ(m)−1

j=0 (x − j) is a null polynomials overZm[α].

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Proposition

Suppose that f (x) = f1(x) + f2(x)α, where f1(x), f2(x) ∈ Z[x ].Then f (x) is a null polynomial over Zm[α] iff f1(x), f ′1(x) andf2(x) are null polynomials modulo m.

Corollary

f (x) = (x)2µ(m) =∏2µ(m)−1

j=0 (x − j) is a null polynomials overZm[α].

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Theorem

Let n ≤ p. For f (x) =∑m

k=0(fk(x)(xp − x)k) ∈ Z[x ],

fk(x) =∑p−1

j=0 ajkxj . Then f (x), f ′(x) are null polynomials modulo

pn iff

aj0 ≡ 0 (mod pn),

ajk ≡ 0 (mod pn−k+1) if 1 ≤ k < n,

ajn ≡

{0 (mod p) if n < p,

0 (mod p0) if n = p,

ajk ≡ 0 (mod p0) if k > n. For 0 ≤ j ≤ p − 1.

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Corollary

Let n ≤ p and f (x) ∈ Z[x ] such that f (x), f ′(x) are nullpolynomials (mod pn) with deg f ≤ (n + 1)p − 1 with coefficientreduced (mod pn). Let N denote the number of all polynomialsf (x).

Then N =

pn(n−1)p

2 if n < p,

p(p2−p+2)p

2 if n = p.

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Polynomial Functions over Zm[α]

Theorem

Let F : Zm[α] −→ Zm[α] defined by F (i + jα) = ci + d(i ,j)α,where ci , d(i ,j) ∈ Zm for i , j = 0, 1, ...,m − 1. T F A E:

F is a polynomial function over Zm[α].

F induced by f (x) =∑2µ−1

k=0 akxk +

∑µ−1l=0 blx

The system of linear congruences,{∑2µ−1k=0 ikxk ≡ ci∑2µ−1k=0 kik−1jxk +

∑µ−1l=0 i lyl ≡ d(i ,j) (mod m)

i , j = 0, 1, ...,m − 1, has a solution xk = ak , yl = bl fork = 0, 1, ..., 2µ− 1, l = 0, 1, ..., µ− 1.

References

Theorem

Let f (x) = f1(x) + f2(x)α, where f1(x), f2(x) ∈ Z[x ]. Then f (x) isa permutation polynomial over Zpn [α] iff f1(x) is a permutationpolynomial (mod p) and f ′1(a) 6≡ 0 for every a ∈ Zp.

References

Let Stabα(Zm) = {F ∈ P(Zm[α]) : F (a) = a for every a ∈ Zm}.

Proposition

Let m = pn11 ...pnkk where p1, .., pk are distinct primes and suppose

that nj > 1 for j = 1, .., k . Then Stabα(Zm) = {F ∈ P(Zm[α]) :F is represented by x + h(x), h(x) ∈ Z[x ] where h(x) is a nullpolynomial modulo m}.

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Let Stabα(Zm) = {F ∈ P(Zm[α]) : F (a) = a for every a ∈ Zm}.

Proposition

Let m = pn11 ...pnkk where p1, .., pk are distinct primes and suppose

that nj > 1 for j = 1, .., k . Then Stabα(Zm) = {F ∈ P(Zm[α]) :F is represented by x + h(x), h(x) ∈ Z[x ] where h(x) is a nullpolynomial modulo m}.

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Counting Formulas

Theorem

Let n > 1. The number of polynomial functions over Zpn [α] isgiven by |F(Zpn [α])| = |F(Zpn)|2 × |Stabα(Zpn)|.

Theorem

Let n > 1. The number of permutation polynomials over Zpn [α] isgiven by |P(Zpn [α])| = |F(Zpn)| × |P(Zpn)| × |Stabα(Zpn)|.

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Counting Formulas

Theorem

Let n > 1. The number of polynomial functions over Zpn [α] isgiven by |F(Zpn [α])| = |F(Zpn)|2 × |Stabα(Zpn)|.

Theorem

Let n > 1. The number of permutation polynomials over Zpn [α] isgiven by |P(Zpn [α])| = |F(Zpn)| × |P(Zpn)| × |Stabα(Zpn)|.

References

Counting Formulas

Proposition

Let 1 < n ≤ p

|Stabα(Zpn)| =

{pnp if n < p,

p(p−1)p if n = p.

References

Counting Formulas

Theorem

For n ≤ p the number of polynomial functions over Zpn [α] is givenby

|F(Zpn [α])| =

2+2n)p if n < p,

p(p2+2p−1)p if n = p.

Corollary

For n ≤ p the number of permutation polynomials over Zpn [α] is

given by |P(Zpn [α])| =

{p!(p − 1)pp(n

2+2n−2)p if n < p,

p!(p − 1)pp(p2+2p−3)p if n = p.

References

Counting Formulas

Theorem

For n ≤ p the number of polynomial functions over Zpn [α] is givenby

|F(Zpn [α])| =

2+2n)p if n < p,

p(p2+2p−1)p if n = p.

Corollary

For n ≤ p the number of permutation polynomials over Zpn [α] is

given by |P(Zpn [α])| =

{p!(p − 1)pp(n

2+2n−2)p if n < p,

p!(p − 1)pp(p2+2p−3)p if n = p.

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Some Generalizations

Theorem

Let Zpn [α1, ..., αk ] = {a + b1α1 + ...+ bkαk : αiαj = 0, a, bi ∈Zpn for i , j = 1, .., k}.

|F(Zpn [α1, ..., αk ])| =

2+2n)ppn(n+1)(k−1)p

2 if n < p,

p(p2+2p−1)pp

n(n+1)(k−1)p2 if n = p.

|P(Zpn [α1, ..., αk ])| ={p!(p − 1)pp(n

2+2n−2)ppn(n+1)(k−1)p

2 if n < p,

p!(p − 1)pp(p2+2p−3)pp

p(p+1)(k−1)p2 if n = p.

References

Theorem

Let Zpn [α1, ..., αk ] = {a + b1α1 + ...+ bkαk : αiαj = 0, a, bi ∈Zpn for i , j = 1, .., k}.Then:

1 For n > 1, |F(Zpn [α1, ..., αk ])| = |F(Zpn)|k+1 × |Stabα(Zpn)|.

2 For n ≤ p,

|F(Zpn [α1, ..., αk ])| =

2+2n)ppn(n+1)(k−1)p

2 if n < p,

p(p2+2p−1)pp

n(n+1)(k−1)p2 if n = p.

|P(Zpn [α1, ..., αk ])| ={p!(p − 1)pp(n

2+2n−2)ppn(n+1)(k−1)p

2 if n < p,

p!(p − 1)pp(p2+2p−3)pp

p(p+1)(k−1)p2 if n = p.

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Theorem

|F(Zpn [α1, ..., αk ])| =

2+2n)ppn(n+1)(k−1)p

2 if n < p,

p(p2+2p−1)pp

n(n+1)(k−1)p2 if n = p.

|P(Zpn [α1, ..., αk ])| ={p!(p − 1)pp(n

2+2n−2)ppn(n+1)(k−1)p

2 if n < p,

p!(p − 1)pp(p2+2p−3)pp

p(p+1)(k−1)p2 if n = p.

References

Theorem

|F(Zpn [α1, ..., αk ])| =

2+2n)ppn(n+1)(k−1)p

2 if n < p,

p(p2+2p−1)pp

n(n+1)(k−1)p2 if n = p.

|P(Zpn [α1, ..., αk ])| ={p!(p − 1)pp(n

2+2n−2)ppn(n+1)(k−1)p

2 if n < p,

p!(p − 1)pp(p2+2p−3)pp

p(p+1)(k−1)p2 if n = p.

References

Chen, Z. (1995). On polynomial functions from Zn to Zm. DiscreteMath., 137(1-3):137–145.

Frisch, S. (1999). Polynomial functions on finite commutativerings. In Advances in commutative ring theory (Fez, 1997),volume 205 of Lecture Notes in Pure and Appl. Math., pages323–336. Dekker, New York.

Frisch, S. and Krenn, D. (2013). Sylow p-groups of polynomialpermutations on the integers mod pn. J. Number Theory,133(12):4188–4199.

Kempner, A. J. (1918). Miscellanea. Amer. Math. Monthly,25(5):201–210.

Kempner, A. J. (1921). Polynomials and their residue systems.Trans. Amer. Math. Soc., 22(2):240–266, 267–288.

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