The solvability cardinal of the class of polynomials

Abstract

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 fin {{mathbb {C}}}^G and xin G. By a system of difference equations we mean a set of equations { D_i f=g_i : iin 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 iin 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 2le 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 2le 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).