Chevalley-Warning type results on abelian groups

Aichinger, Erhard; Moosbauer, Jakob

doi:10.1016/j.jalgebra.2020.10.033

Cited by 20 publications

(42 citation statements)

References 20 publications

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“…for all a ∈ m i=1 A i . Furthermore, as in [3], we can see that the component ) ). From now on we will not specify the element that the functions absorb since it will always be the 0 of a finite field.…”

Section: Preliminaries and Notationsupporting

confidence: 56%

“…We denote by R[M ] the monoid ring of M over R. Following the notation in [3] for all a ∈ A we define τ a to be the element of R M with τ a (a) = 1 and τ a (M \{a}) = {0}. We observe that for all f ∈ R[M ] there is an r ∈ R M such that f = a∈M r a τ a and that we can multiply such expressions with the rule τ a • τ b = τ ab .…”

Section: The Lattice Of All (F K)-linearly Closed Clonoidsmentioning

confidence: 99%

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Closed sets of finitary functions between products of finite fields of coprime order

Fioravanti

2021

Algebra Univers.

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We investigate the finitary functions from a finite product of finite fields $$\prod _{j =1}^m\mathbb {F}_{q_j} = {\mathbb K}$$ ∏ j = 1 m F q j = K to a finite product of finite fields $$\prod _{i =1}^n\mathbb {F}_{p_i} = {\mathbb {F}}$$ ∏ i = 1 n F p i = F , where $$|{\mathbb K}|$$ | K | and $$|{\mathbb {F}}|$$ | F | are coprime. An $$({\mathbb {F}},{\mathbb K})$$ ( F , K ) -linearly closed clonoid is a subset of these functions which is closed under composition from the right and from the left with linear mappings. We give a characterization of these subsets of functions through the $${\mathbb {F}}_p[{\mathbb K}^{\times }]$$ F p [ K × ] -submodules of $$\mathbb {F}_p^{{\mathbb K}}$$ F p K , where $${\mathbb K}^{\times }$$ K × is the multiplicative monoid of $${\mathbb K}= \prod _{i=1}^m {\mathbb {F}}_{q_i}$$ K = ∏ i = 1 m F q i . Furthermore we prove that each of these subsets of functions is generated by a set of unary functions and we provide an upper bound for the number of distinct $$({\mathbb {F}},{\mathbb K})$$ ( F , K ) -linearly closed clonoids.

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Section: Preliminaries and Notationsupporting

confidence: 56%

Section: The Lattice Of All (F K)-linearly Closed Clonoidsmentioning

confidence: 99%

Closed sets of finitary functions between products of finite fields of coprime order

Fioravanti

2021

Algebra Univers.

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“…Things become easier if we consider Fréchet's mixed differences functional equation, which is, in quite general cases, equivalent to the unmixed equation (see, e.g., [2], [3], [4], [5], [6], [7], [10], [12], [15]). The proof recently appeared in a paper by Aichinger and Moosbauer [1]. In their paper, the authors introduced the following wonderful functional equation, that characterizes generalized polynomials of degree ď m on abelian groups: (1) f px 1 `¨¨¨`x m`1 q " m`1 ÿ i"1 g i px 1 , x 2 , ¨¨¨, p x i , ¨¨¨, x m`1 q, where p x i means that the function g i does not depend on x i , and they used this equation to prove that the composition f ˝g of the generalized polynomials (defined on abelian groups) f , g is a generalized polynomial and degpf ˝gq ď degpf q ¨degpgq.…”

Section: Introductionmentioning

confidence: 97%

“…The proof recently appeared in a paper by Aichinger and Moosbauer [1]. In their paper, the authors introduced the following wonderful functional equation, that characterizes generalized polynomials of degree ď m on abelian groups: (1) f px 1 `¨¨¨`x m`1 q " m`1 ÿ i"1 g i px 1 , x 2 , ¨¨¨, p x i , ¨¨¨, x m`1 q, where p x i means that the function g i does not depend on x i , and they used this equation to prove that the composition f ˝g of the generalized polynomials (defined on abelian groups) f , g is a generalized polynomial and degpf ˝gq ď degpf q ¨degpgq. This same result, with a different proof that does not use (1), had already been proved by Leibman in [16], but the authors of [1] were unaware of that paper.…”

Section: Introductionmentioning

confidence: 97%

“…In their paper, the authors introduced the following wonderful functional equation, that characterizes generalized polynomials of degree ď m on abelian groups: (1) f px 1 `¨¨¨`x m`1 q " m`1 ÿ i"1 g i px 1 , x 2 , ¨¨¨, p x i , ¨¨¨, x m`1 q, where p x i means that the function g i does not depend on x i , and they used this equation to prove that the composition f ˝g of the generalized polynomials (defined on abelian groups) f , g is a generalized polynomial and degpf ˝gq ď degpf q ¨degpgq. This same result, with a different proof that does not use (1), had already been proved by Leibman in [16], but the authors of [1] were unaware of that paper. Moreover in his paper Leibman also proved that if we consider a composition of several polynomial functions f i : G i´1 Ñ G i , i " 1, 2, ..., k, and G k is nilpotent (with no other extra hypotheses on the first groups G 0 , G 1 , ..., G k´1 , that may be noncommutative and quite general, indeed), the composition f k ˝fk´1 ˝¨¨¨˝f 1 is also a polynomial function, but no precise estimation of its degree was given.…”

Section: Introductionmentioning

confidence: 99%

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Aichinger equation on commutative semigroups

Almira¹

2022

Preprint

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We consider Aichinger's equation f px 1 `¨¨¨`x m`1 q " m`1 ÿ i"1for functions defined on commutative semigroups which take values on commutative groups. The solutions of this equation are, under very mild hypotheses, generalized polynomials. We use the canonical form of generalized polynomials to prove that compositions and products of generalized polynomials are again generalized polynomials and that the bounds for the degrees are, in this new context, the natural ones. In some cases, we also show that a polynomial function defined on a semigroup can uniquely be extended to a polynomial function defined on a larger group. For example, if f solves Aichinger's equation under the additional restriction that x 1 , ¨¨¨, x m`1 P R p `, then there exists a unique polynomial function F defined on R p such that F |R p `" f . In particular, if f is also bounded on a set A Ď R p `with positive Lebesgue measure then its unique polynomial extension F is an ordinary polynomial of p variables with total degree ď m, and the functions g i are also restrictions to R pm `of ordinary polynomials of total degree ď m defined on R pm .

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Using Aichinger’s equation to characterize polynomial functions

Almira

2022

Aequat. Math.

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Aichinger’s equation is used to give simple proofs of several well-known characterizations of polynomial functions as solutions of certain functional equations. Concretely, we use that Aichinger’s equation characterizes polynomial functions to solve, for arbitrary commutative groups, Ghurye–Olkin’s functional equation, Wilson’s functional equation, the Kakutani–Nagumo–Walsh functional equation, and a general version of Fréchet’s unmixed functional equation.

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Chevalley-Warning type results on abelian groups

Cited by 20 publications

References 20 publications

Closed sets of finitary functions between products of finite fields of coprime order

Closed sets of finitary functions between products of finite fields of coprime order

Aichinger equation on commutative semigroups

Using Aichinger’s equation to characterize polynomial functions

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