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

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Background Dual Numbers Null polynomials over Zm[α] Polynomial Functions over Zm[α] Counting Formulas Some Generalizations References Polynomial Functions of the Ring of Dual Numbers Modulo m Amr Al-Maktry 1 Hasan Al-Ezeh 2 Sophie Frisch 1 1 TU Graz University 2 Jordan 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

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

Page 1: Polynomial Functions of the Ring of Dual Numbers Modulo · Background Dual Numbers Null polynomials over Zm[ ] Polynomial Functions over Zm[ ] Counting Formulas Some Generalizations

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

Page 2: Polynomial Functions of the Ring of Dual Numbers Modulo · Background Dual Numbers Null polynomials over Zm[ ] Polynomial Functions over Zm[ ] Counting Formulas Some Generalizations

BackgroundDual Numbers

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

Counting FormulasSome Generalizations

References

Outline

1 Background

2 Dual Numbers

3 Null polynomials over Zm[α]

4 Polynomial Functions over Zm[α]

5 Counting Formulas

6 Some Generalizations

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

Page 3: Polynomial Functions of the Ring of Dual Numbers Modulo · Background Dual Numbers Null polynomials over Zm[ ] Polynomial Functions over Zm[ ] Counting Formulas Some Generalizations

BackgroundDual Numbers

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

Counting FormulasSome 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).

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

Page 4: Polynomial Functions of the Ring of Dual Numbers Modulo · Background Dual Numbers Null polynomials over Zm[ ] Polynomial Functions over Zm[ ] Counting Formulas Some Generalizations

BackgroundDual Numbers

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

Counting FormulasSome 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).

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

Page 5: Polynomial Functions of the Ring of Dual Numbers Modulo · Background Dual Numbers Null polynomials over Zm[ ] Polynomial Functions over Zm[ ] Counting Formulas Some Generalizations

BackgroundDual Numbers

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

Counting FormulasSome 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).

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

Page 6: Polynomial Functions of the Ring of Dual Numbers Modulo · Background Dual Numbers Null polynomials over Zm[ ] Polynomial Functions over Zm[ ] Counting Formulas Some Generalizations

BackgroundDual Numbers

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

Counting FormulasSome 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).

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

Page 7: Polynomial Functions of the Ring of Dual Numbers Modulo · Background Dual Numbers Null polynomials over Zm[ ] Polynomial Functions over Zm[ ] Counting Formulas Some Generalizations

BackgroundDual Numbers

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

Counting FormulasSome 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).

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

Page 8: Polynomial Functions of the Ring of Dual Numbers Modulo · Background Dual Numbers Null polynomials over Zm[ ] Polynomial Functions over Zm[ ] Counting Formulas Some Generalizations

BackgroundDual Numbers

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

Counting FormulasSome 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).

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

Page 9: Polynomial Functions of the Ring of Dual Numbers Modulo · Background Dual Numbers Null polynomials over Zm[ ] Polynomial Functions over Zm[ ] Counting Formulas Some Generalizations

BackgroundDual Numbers

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

Counting FormulasSome Generalizations

References

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).

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

Page 10: Polynomial Functions of the Ring of Dual Numbers Modulo · Background Dual Numbers Null polynomials over Zm[ ] Polynomial Functions over Zm[ ] Counting Formulas Some Generalizations

BackgroundDual Numbers

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

Counting FormulasSome Generalizations

References

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).

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

Page 11: Polynomial Functions of the Ring of Dual Numbers Modulo · Background Dual Numbers Null polynomials over Zm[ ] Polynomial Functions over Zm[ ] Counting Formulas Some Generalizations

BackgroundDual Numbers

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

Counting FormulasSome Generalizations

References

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).

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

Page 12: Polynomial Functions of the Ring of Dual Numbers Modulo · Background Dual Numbers Null polynomials over Zm[ ] Polynomial Functions over Zm[ ] Counting Formulas Some Generalizations

BackgroundDual Numbers

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

Counting FormulasSome Generalizations

References

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).

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

Page 13: Polynomial Functions of the Ring of Dual Numbers Modulo · Background Dual Numbers Null polynomials over Zm[ ] Polynomial Functions over Zm[ ] Counting Formulas Some Generalizations

BackgroundDual Numbers

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

Counting FormulasSome Generalizations

References

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).

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

Page 14: Polynomial Functions of the Ring of Dual Numbers Modulo · Background Dual Numbers Null polynomials over Zm[ ] Polynomial Functions over Zm[ ] Counting Formulas Some Generalizations

BackgroundDual Numbers

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

Counting FormulasSome Generalizations

References

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}

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

Page 15: Polynomial Functions of the Ring of Dual Numbers Modulo · Background Dual Numbers Null polynomials over Zm[ ] Polynomial Functions over Zm[ ] Counting Formulas Some Generalizations

BackgroundDual Numbers

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

Counting FormulasSome Generalizations

References

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}

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

Page 16: Polynomial Functions of the Ring of Dual Numbers Modulo · Background Dual Numbers Null polynomials over Zm[ ] Polynomial Functions over Zm[ ] Counting Formulas Some Generalizations

BackgroundDual Numbers

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

Counting FormulasSome Generalizations

References

Fact

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

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

Page 17: Polynomial Functions of the Ring of Dual Numbers Modulo · Background Dual Numbers Null polynomials over Zm[ ] Polynomial Functions over Zm[ ] Counting Formulas Some Generalizations

BackgroundDual Numbers

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

Counting FormulasSome Generalizations

References

Fact

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

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

Page 18: Polynomial Functions of the Ring of Dual Numbers Modulo · Background Dual Numbers Null polynomials over Zm[ ] Polynomial Functions over Zm[ ] Counting Formulas Some Generalizations

BackgroundDual Numbers

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

Counting FormulasSome Generalizations

References

Fact

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

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

Page 19: Polynomial Functions of the Ring of Dual Numbers Modulo · Background Dual Numbers Null polynomials over Zm[ ] Polynomial Functions over Zm[ ] Counting Formulas Some Generalizations

BackgroundDual Numbers

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

Counting FormulasSome Generalizations

References

Fact

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

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

Page 20: Polynomial Functions of the Ring of Dual Numbers Modulo · Background Dual Numbers Null polynomials over Zm[ ] Polynomial Functions over Zm[ ] Counting Formulas Some Generalizations

BackgroundDual Numbers

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

Counting FormulasSome Generalizations

References

Fact

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

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

Page 21: Polynomial Functions of the Ring of Dual Numbers Modulo · Background Dual Numbers Null polynomials over Zm[ ] Polynomial Functions over Zm[ ] Counting Formulas Some Generalizations

BackgroundDual Numbers

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

Counting FormulasSome Generalizations

References

Fact

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

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

Page 22: Polynomial Functions of the Ring of Dual Numbers Modulo · Background Dual Numbers Null polynomials over Zm[ ] Polynomial Functions over Zm[ ] Counting Formulas Some Generalizations

BackgroundDual Numbers

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

Counting FormulasSome Generalizations

References

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.

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

Page 23: Polynomial Functions of the Ring of Dual Numbers Modulo · Background Dual Numbers Null polynomials over Zm[ ] Polynomial Functions over Zm[ ] Counting Formulas Some Generalizations

BackgroundDual Numbers

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

Counting FormulasSome Generalizations

References

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.

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

Page 24: Polynomial Functions of the Ring of Dual Numbers Modulo · Background Dual Numbers Null polynomials over Zm[ ] Polynomial Functions over Zm[ ] Counting Formulas Some Generalizations

BackgroundDual Numbers

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

Counting FormulasSome Generalizations

References

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.

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

Page 25: Polynomial Functions of the Ring of Dual Numbers Modulo · Background Dual Numbers Null polynomials over Zm[ ] Polynomial Functions over Zm[ ] Counting Formulas Some Generalizations

BackgroundDual Numbers

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

Counting FormulasSome Generalizations

References

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].

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

Page 26: Polynomial Functions of the Ring of Dual Numbers Modulo · Background Dual Numbers Null polynomials over Zm[ ] Polynomial Functions over Zm[ ] Counting Formulas Some Generalizations

BackgroundDual Numbers

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

Counting FormulasSome Generalizations

References

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].

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

Page 27: Polynomial Functions of the Ring of Dual Numbers Modulo · Background Dual Numbers Null polynomials over Zm[ ] Polynomial Functions over Zm[ ] Counting Formulas Some Generalizations

BackgroundDual Numbers

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

Counting FormulasSome Generalizations

References

Null polynomials over Zm[α]

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.

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

Page 28: Polynomial Functions of the Ring of Dual Numbers Modulo · Background Dual Numbers Null polynomials over Zm[ ] Polynomial Functions over Zm[ ] Counting Formulas Some Generalizations

BackgroundDual Numbers

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

Counting FormulasSome Generalizations

References

Null polynomials over Zm[α]

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.

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

Page 29: Polynomial Functions of the Ring of Dual Numbers Modulo · Background Dual Numbers Null polynomials over Zm[ ] Polynomial Functions over Zm[ ] Counting Formulas Some Generalizations

BackgroundDual Numbers

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

Counting FormulasSome Generalizations

References

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

lα.

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.

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

Page 30: Polynomial Functions of the Ring of Dual Numbers Modulo · Background Dual Numbers Null polynomials over Zm[ ] Polynomial Functions over Zm[ ] Counting Formulas Some Generalizations

BackgroundDual Numbers

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

Counting FormulasSome Generalizations

References

Polynomial Functions over Zm[α]

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.

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

Page 31: Polynomial Functions of the Ring of Dual Numbers Modulo · Background Dual Numbers Null polynomials over Zm[ ] Polynomial Functions over Zm[ ] Counting Formulas Some Generalizations

BackgroundDual Numbers

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

Counting FormulasSome Generalizations

References

Polynomial Functions over Zm[α]

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}.

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

Page 32: Polynomial Functions of the Ring of Dual Numbers Modulo · Background Dual Numbers Null polynomials over Zm[ ] Polynomial Functions over Zm[ ] Counting Formulas Some Generalizations

BackgroundDual Numbers

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

Counting FormulasSome Generalizations

References

Polynomial Functions over Zm[α]

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)|.

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Counting Formulas

Proposition

Let 1 < n ≤ p

|Stabα(Zpn)| =

{pnp if n < p,

p(p−1)p if n = p.

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

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Counting Formulas

Theorem

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

|F(Zpn [α])| =

{p(n

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.

.

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

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Counting Formulas

Theorem

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

|F(Zpn [α])| =

{p(n

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.

.

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

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Some Generalizations

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 ])| =

{p(n

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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Some Generalizations

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 ])| =

{p(n

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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Some Generalizations

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 ])| =

{p(n

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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Some Generalizations

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 ])| =

{p(n

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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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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