论文标题
通勤半群的Aichinger方程
Aichinger equation on commutative semigroups
论文作者
论文摘要
我们考虑aichinger的方程$$ f（x_1+\ cdots+x_ {m+1}）= \ sum_ {i = 1}^{m+1} g_i（x_1，x_1，x_2，\ cdots，\ cdots，\ wideHat {x_i}组。在非常温和的假设下，该方程的解决方案是普遍的多项式。我们使用广义多项式的规范形式来证明广义多项式的组成和产物再次是广义的多项式，并且在这种新的情况下，该程度的界限是自然的。在某些情况下，我们还表明，在半群上定义的多项式函数可以唯一地扩展到在较大组上定义的多项式函数。例如，如果$ f $在$ x_1，\ cdots，x_ {m+1} \ in \ mathbb {r} _+^p $的附加限制下求解了aichinger的方程$ f_ {| \ mathbb {r} _+^p} = f $。特别是，如果$ f $在一组$ a \ subseteq \ mathbb {r} _+^p $带有积极的lebesgue量度的限制下，则其唯一的多项式扩展$ f $是$ p $ $ p $变量的普通多项式变量，具有总数$ \ leq m $，并且功能$ g_i $ $ g_i $^$^r^r^r^r^r}在$ \ mathbb {r}^{pm} $上定义的总数$ \ leq m $的普通多项式。
We consider Aichinger's equation $$f(x_1+\cdots+x_{m+1})=\sum_{i=1}^{m+1}g_i(x_1,x_2,\cdots, \widehat{x_i},\cdots, x_{m+1})$$ for 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 $x_1,\cdots,x_{m+1}\in \mathbb{R}_+^p$, then there exists a unique polynomial function $F$ defined on $\mathbb{R}^p$ such that $F_{|\mathbb{R}_+^p}=f$. In particular, if $f$ is also bounded on a set $A\subseteq \mathbb{R}_+^p$ with positive Lebesgue measure then its unique polynomial extension $F$ is an ordinary polynomial of $p$ variables with total degree $\leq m$, and the functions $g_i$ are also restrictions to $\mathbb{R}_+^{pm}$ of ordinary polynomials of total degree $\leq m$ defined on $\mathbb{R}^{pm}$.
