Let G be an Abelian group, and let \({{\mathbb {C}}}^G\) denote the set of complex valued functions defined on G. A map \(D: {{\mathbb {C}}}^G \rightarrow {{\mathbb {C}}}^G\) is a difference operator, if there are complex numbers \(a_i\) and elements \(b_i \in G\) \((i=1,\ldots , n)\) such that \((Df)(x)=\sum _{i=1}^n a_i f(x+b_i)\) for every \(f\in {{\mathbb {C}}}^G \) and \(x\in G\). By a system of difference equations we mean a set of equations \(\{ D_i f=g_i : i\in I\}\), where I is an arbitrary set of indices, \(D_i\) is a difference operator and \(g_i \in {{\mathbb {C}}}^G\) is a given function for every \(i\in I\), and f is the unknown function. The solvability cardinal \(\mathrm{sc} \,({{\mathcal {F}}})\) of a class of functions \({{\mathcal {F}}} \subset {{\mathbb {C}}}^G\) is the smallest cardinal number \(\kappa \) with the following property: whenever S is a system of difference equations on G such that each subsystem of S of cardinality \(<\kappa \) has a solution in \({{\mathcal {F}}}\), then S itself has a solution in \({{\mathcal {F}}}\). The behaviour of \(\mathrm{sc} \,({{\mathcal {F}}})\) is rather erratic, even for classes of functions defined on \({{\mathbb {R}}}\). For example, \(\mathrm{sc} \,({{\mathbb {C}}}[x])=3\), but \(\mathrm{sc} \,({\mathcal {TP}}) =\omega _1\), where \({\mathcal {TP}}\) is the set of trigonometric polynomials; \(\mathrm{sc} \,({{\mathbb {C}}}^{{\mathbb {R}}})=\omega \), but \(\mathrm{sc} \,({\mathcal {DF}}) =(2^\omega )^+\), where \({\mathcal {DF}}\) is the set of functions having the Darboux property. Our aim is to determine or to estimate the solvability cardinal of the class of polynomials defined on \({{{\mathbb {R}}}}^n\), on normed linear spaces and, in general, on topological Abelian groups. Let \({{\mathcal {P}}}_G\) denote the class of polynomials defined on the group G. After presenting some general estimates we prove that \(\mathrm{sc} \,({{\mathbb {C}}}[x_1 ,\ldots ,x_n ])=\omega \) if \(2\le n<\infty \), and \(\mathrm{sc} \,({{\mathcal {P}}}_X)=\omega _1\) if X is a normed linear space of infinite dimension. For discrete Abelian groups we show that \(\mathrm{sc} \,({{\mathcal {P}}}_G)=3\) if \(r_0 (G)\le 1\), \(\mathrm{sc} \,({{\mathcal {P}}}_G)=\omega \) if \(2\le r_0 (G)<\infty \), and \(\mathrm{sc} \,({{\mathcal {P}}}_G)\ge \omega _1\) if \(r_0 (G)\) is infinite, where \(r_0 (G)\) denotes the torsion free rank of G. The solvability of systems of difference equations is closely connected to the existence of projections of function classes commuting with translations (see Theorem 7.1). As an application we construct a projection from \({{\mathbb {C}}}^{{{{\mathbb {R}}}}^n}\) onto \({{\mathbb {C}}}[x_1 ,\ldots ,x_n ]\) commuting with translations by vectors having rational coordinates (Theorem 7.4).

令G为阿贝尔群，并令\({{\mathbb {C}}}^G\)表示在G 上定义的复数值函数集。映射\(D: {{\mathbb {C}}}^G \rightarrow {{\mathbb {C}}}^G\)是差分算子，如果有复数\(a_i\)和元素\ (b_i \in G\) \((i=1,\ldots , n)\)使得\((Df)(x)=\sum _{i=1}^n a_i f(x+b_i)\ )对于每个\(f\in {{\mathbb {C}}}^G \)和\(x\in G\)。差分方程组是指一组方程\(\{ D_i f=g_i : i\in I\}\)，其中I是一组任意的索引，\(D_i\)是一个差分运算符，\(g_i \in {{\mathbb {C}}}^G\)是每个\(i\in I\)的给定函数，而f是未知函数。一类函数的可解性基数\(\mathrm{sc} \,({{\mathcal {F}}})\) \({{\mathcal {F}}} \subset {{\mathbb {C} }}^G\)是最小基数\(\kappa \)具有以下性质：只要S是G上的差分方程组，使得S的基数\(<\kappa \) 的每个子系统都有解在\({{\mathcal {F}}}\)，然后S本身有一个解决方案\({{\mathcal {F}}}\)。的行为\（\ mathrm {SC} \，（{{\ mathcal {F}}}）\）是相当不稳定的，即使对于上定义的函数的类\（{{\ mathbb {R}}} \） 。例如，\(\mathrm{sc} \,({{\mathbb {C}}}[x])=3\)，但是\(\mathrm{sc} \,({\mathcal {TP}}) =\omega _1\)，其中\({\mathcal {TP}}\)是三角多项式的集合；\(\mathrm{sc} \,({{\mathbb {C}}}^{{\mathbb {R}}})=\omega \)，但是\(\mathrm{sc} \,({\mathcal {DF}}) =(2^\omega )^+\)，其中\({\mathcal {DF}}\)是具有 Darboux 性质的函数集。我们的目标是确定或估计定义在\({{{\mathbb {R}}}}^n\)、赋范线性空间和一般拓扑阿贝尔群上的多项式类的可解性基数。让\({{\mathcal {P}}}_G\)表示定义在群G上的多项式类。在提出一些一般估计之后，我们证明\(\mathrm{sc} \,({{\mathbb {C}}}[x_1 ,\ldots ,x_n ])=\omega \)如果\(2\le n<\ infty \)和\(\mathrm{sc} \,({{\mathcal {P}}}_X)=\omega _1\)如果X是无限维的赋范线性空间。对于离散阿贝尔群，我们证明\(\mathrm{sc} \,({{\mathcal {P}}}_G)=3\) if \(r_0 (G)\le 1\) , \(\mathrm{sc} \,({{ \mathcal {P}}}_G)=\omega \)如果\(2\le r_0 (G)<\infty \)和\(\mathrm{sc} \,({{\mathcal {P}}} _G）\ GE \欧米加_1 \）如果\（R_0（G）\）是无限的，其中\（R_0（G）\）表示的无扭转的秩ģ。差分方程组的可解性与具有平移交换的函数类的投影的存在密切相关（见定理 7.1）。作为一个应用程序，我们构建了一个从\({{\mathbb {C}}}}^{{{\mathbb {R}}}}^n}\)到\({{\mathbb {C}}}[ x_1 ,\ldots ,x_n ]\) 通过具有有理坐标的向量进行平移（定理 7.4）。