The solvability cardinal of the class of polynomials

Abstract

AbstractLet G be an Abelian group, and let $${{\mathbb {C}}}^G$$ C G denote the set of complex valued functions defined on G. A map $$D: {{\mathbb {C}}}^G \rightarrow {{\mathbb {C}}}^G$$ D : C G → C G is a difference operator, if there are complex numbers $$a_i$$ a i and elements $$b_i \in G$$ b i ∈ G $$(i=1,\ldots , n)$$ ( i = 1 , … , n ) such that $$(Df)(x)=\sum _{i=1}^n a_i f(x+b_i)$$ ( D f ) ( x ) = ∑ i = 1 n a i f ( x + b i ) for every $$f\in {{\mathbb {C}}}^G $$ f ∈ C G and $$x\in G$$ x ∈ G . By a system of difference equations we mean a set of equations $$\{ D_i f=g_i : i\in I\}$$ { D i f = g i : i ∈ I } , where I is an arbitrary set of indices, $$D_i$$ D i is a difference operator and $$g_i \in {{\mathbb {C}}}^G$$ g i ∈ C G is a given function for every $$i\in I$$ i ∈ I , and f is the unknown function. The solvability cardinal $$\mathrm{sc} \,({{\mathcal {F}}})$$ sc ( F ) of a class of functions $${{\mathcal {F}}} \subset {{\mathbb {C}}}^G$$ F ⊂ C G is the smallest cardinal number $$\kappa $$ κ with the following property: whenever S is a system of difference equations on G such that each subsystem of S of cardinality $$<\kappa $$ < κ has a solution in $${{\mathcal {F}}}$$ F , then S itself has a solution in $${{\mathcal {F}}}$$ F . The behaviour of $$\mathrm{sc} \,({{\mathcal {F}}})$$ sc ( F ) is rather erratic, even for classes of functions defined on $${{\mathbb {R}}}$$ R . For example, $$\mathrm{sc} \,({{\mathbb {C}}}[x])=3$$ sc ( C [ x ] ) = 3 , but $$\mathrm{sc} \,({\mathcal {TP}}) =\omega _1$$ sc ( TP ) = ω 1 , where $${\mathcal {TP}}$$ TP is the set of trigonometric polynomials; $$\mathrm{sc} \,({{\mathbb {C}}}^{{\mathbb {R}}})=\omega $$ sc ( C R ) = ω , but $$\mathrm{sc} \,({\mathcal {DF}}) =(2^\omega )^+$$ sc ( DF ) = ( 2 ω ) + , where $${\mathcal {DF}}$$ 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 $${{{\mathbb {R}}}}^n$$ R n , on normed linear spaces and, in general, on topological Abelian groups. Let $${{\mathcal {P}}}_G$$ P G denote the class of polynomials defined on the group G. After presenting some general estimates we prove that $$\mathrm{sc} \,({{\mathbb {C}}}[x_1 ,\ldots ,x_n ])=\omega $$ sc ( C [ x 1 , … , x n ] ) = ω if $$2\le n<\infty $$ 2 ≤ n < ∞ , and $$\mathrm{sc} \,({{\mathcal {P}}}_X)=\omega _1$$ sc ( P X ) = ω 1 if X is a normed linear space of infinite dimension. For discrete Abelian groups we show that $$\mathrm{sc} \,({{\mathcal {P}}}_G)=3$$ sc ( P G ) = 3 if $$r_0 (G)\le 1$$ r 0 ( G ) ≤ 1 , $$\mathrm{sc} \,({{\mathcal {P}}}_G)=\omega $$ sc ( P G ) = ω if $$2\le r_0 (G)<\infty $$ 2 ≤ r 0 ( G ) < ∞ , and $$\mathrm{sc} \,({{\mathcal {P}}}_G)\ge \omega _1$$ sc ( P G ) ≥ ω 1 if $$r_0 (G)$$ r 0 ( G ) is infinite, where $$r_0 (G)$$ 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 $${{\mathbb {C}}}^{{{{\mathbb {R}}}}^n}$$ C R n onto $${{\mathbb {C}}}[x_1 ,\ldots ,x_n ]$$ C [ x 1 , … , x n ] commuting with translations by vectors having rational coordinates (Theorem 7.4).