![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](https://reader034.vdocuments.pub/reader034/viewer/2022042100/5e7cfd09f3820661ac7d62d9/html5/thumbnails/1.jpg)
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](https://reader034.vdocuments.pub/reader034/viewer/2022042100/5e7cfd09f3820661ac7d62d9/html5/thumbnails/2.jpg)
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](https://reader034.vdocuments.pub/reader034/viewer/2022042100/5e7cfd09f3820661ac7d62d9/html5/thumbnails/3.jpg)
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](https://reader034.vdocuments.pub/reader034/viewer/2022042100/5e7cfd09f3820661ac7d62d9/html5/thumbnails/4.jpg)
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](https://reader034.vdocuments.pub/reader034/viewer/2022042100/5e7cfd09f3820661ac7d62d9/html5/thumbnails/5.jpg)
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](https://reader034.vdocuments.pub/reader034/viewer/2022042100/5e7cfd09f3820661ac7d62d9/html5/thumbnails/6.jpg)
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](https://reader034.vdocuments.pub/reader034/viewer/2022042100/5e7cfd09f3820661ac7d62d9/html5/thumbnails/7.jpg)
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](https://reader034.vdocuments.pub/reader034/viewer/2022042100/5e7cfd09f3820661ac7d62d9/html5/thumbnails/8.jpg)
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](https://reader034.vdocuments.pub/reader034/viewer/2022042100/5e7cfd09f3820661ac7d62d9/html5/thumbnails/9.jpg)
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](https://reader034.vdocuments.pub/reader034/viewer/2022042100/5e7cfd09f3820661ac7d62d9/html5/thumbnails/10.jpg)
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](https://reader034.vdocuments.pub/reader034/viewer/2022042100/5e7cfd09f3820661ac7d62d9/html5/thumbnails/11.jpg)
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](https://reader034.vdocuments.pub/reader034/viewer/2022042100/5e7cfd09f3820661ac7d62d9/html5/thumbnails/12.jpg)
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](https://reader034.vdocuments.pub/reader034/viewer/2022042100/5e7cfd09f3820661ac7d62d9/html5/thumbnails/13.jpg)
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](https://reader034.vdocuments.pub/reader034/viewer/2022042100/5e7cfd09f3820661ac7d62d9/html5/thumbnails/14.jpg)
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](https://reader034.vdocuments.pub/reader034/viewer/2022042100/5e7cfd09f3820661ac7d62d9/html5/thumbnails/15.jpg)
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](https://reader034.vdocuments.pub/reader034/viewer/2022042100/5e7cfd09f3820661ac7d62d9/html5/thumbnails/16.jpg)
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](https://reader034.vdocuments.pub/reader034/viewer/2022042100/5e7cfd09f3820661ac7d62d9/html5/thumbnails/17.jpg)
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](https://reader034.vdocuments.pub/reader034/viewer/2022042100/5e7cfd09f3820661ac7d62d9/html5/thumbnails/18.jpg)
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](https://reader034.vdocuments.pub/reader034/viewer/2022042100/5e7cfd09f3820661ac7d62d9/html5/thumbnails/19.jpg)
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](https://reader034.vdocuments.pub/reader034/viewer/2022042100/5e7cfd09f3820661ac7d62d9/html5/thumbnails/20.jpg)
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](https://reader034.vdocuments.pub/reader034/viewer/2022042100/5e7cfd09f3820661ac7d62d9/html5/thumbnails/21.jpg)
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](https://reader034.vdocuments.pub/reader034/viewer/2022042100/5e7cfd09f3820661ac7d62d9/html5/thumbnails/22.jpg)
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](https://reader034.vdocuments.pub/reader034/viewer/2022042100/5e7cfd09f3820661ac7d62d9/html5/thumbnails/23.jpg)
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](https://reader034.vdocuments.pub/reader034/viewer/2022042100/5e7cfd09f3820661ac7d62d9/html5/thumbnails/24.jpg)
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](https://reader034.vdocuments.pub/reader034/viewer/2022042100/5e7cfd09f3820661ac7d62d9/html5/thumbnails/25.jpg)
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](https://reader034.vdocuments.pub/reader034/viewer/2022042100/5e7cfd09f3820661ac7d62d9/html5/thumbnails/26.jpg)
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](https://reader034.vdocuments.pub/reader034/viewer/2022042100/5e7cfd09f3820661ac7d62d9/html5/thumbnails/27.jpg)
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
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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
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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
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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
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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
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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
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BackgroundDual Numbers
Null polynomials over Zm [α]Polynomial Functions over Zm [α]
Counting FormulasSome Generalizations
References
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)|.
Amr Al-Maktry, Hasan Al-Ezeh, Sophie Frisch Polynomial Functions of the Ring of Dual Numbers Modulo m
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BackgroundDual Numbers
Null polynomials over Zm [α]Polynomial Functions over Zm [α]
Counting FormulasSome Generalizations
References
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)|.
Amr Al-Maktry, Hasan Al-Ezeh, Sophie Frisch Polynomial Functions of the Ring of Dual Numbers Modulo m
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BackgroundDual Numbers
Null polynomials over Zm [α]Polynomial Functions over Zm [α]
Counting FormulasSome Generalizations
References
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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BackgroundDual Numbers
Null polynomials over Zm [α]Polynomial Functions over Zm [α]
Counting FormulasSome Generalizations
References
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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BackgroundDual Numbers
Null polynomials over Zm [α]Polynomial Functions over Zm [α]
Counting FormulasSome Generalizations
References
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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BackgroundDual Numbers
Null polynomials over Zm [α]Polynomial Functions over Zm [α]
Counting FormulasSome Generalizations
References
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.
Amr Al-Maktry, Hasan Al-Ezeh, Sophie Frisch Polynomial Functions of the Ring of Dual Numbers Modulo m
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BackgroundDual Numbers
Null polynomials over Zm [α]Polynomial Functions over Zm [α]
Counting FormulasSome Generalizations
References
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.
Amr Al-Maktry, Hasan Al-Ezeh, Sophie Frisch Polynomial Functions of the Ring of Dual Numbers Modulo m
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BackgroundDual Numbers
Null polynomials over Zm [α]Polynomial Functions over Zm [α]
Counting FormulasSome Generalizations
References
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.
Amr Al-Maktry, Hasan Al-Ezeh, Sophie Frisch Polynomial Functions of the Ring of Dual Numbers Modulo m
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BackgroundDual Numbers
Null polynomials over Zm [α]Polynomial Functions over Zm [α]
Counting FormulasSome Generalizations
References
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.
Amr Al-Maktry, Hasan Al-Ezeh, Sophie Frisch Polynomial Functions of the Ring of Dual Numbers Modulo m
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BackgroundDual Numbers
Null polynomials over Zm [α]Polynomial Functions over Zm [α]
Counting FormulasSome Generalizations
References
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.
Amr Al-Maktry, Hasan Al-Ezeh, Sophie Frisch Polynomial Functions of the Ring of Dual Numbers Modulo m