We prove that for every $N e 4$ there is only one right triangle that tiles the regular $N$-gon.
Let $G$ be a topological Abelian semigroup with unit, let $E$ be a Banach space, and let $C(G,E)$ denote the set of continuous functions $fcolon Gto E$. A function $fin C(G,E)$ is a generalized polynomial, if there is an $nge 0$ such that $Delta_{h_1
} ldots Delta_{h_{n+1}} f=0$ for every $h_1 ,ldots , h_{n+1} in G$, where $Delta_h$ is the difference operator. We say that $fin C(G,E)$ is a polynomial, if it is a generalized polynomial, and the linear span of its translates is of finite dimension; $f$ is a w-polynomial, if $ucirc f$ is a polynomial for every $uin E^*$, and $f$ is a local polynomial, if it is a polynomial on every finitely generated subsemigroup. We show that each of the classes of polynomials, w-polynomials, generalized polynomials, local polynomials is contained in the next class. If $G$ is an Abelian group and has a dense subgroup with finite torsion free rank, then these classes coincide. We introduce the classes of exponential polynomials and w-expo-nential polynomials as well, establish their representations and connection with polynomials and w-polynomials. We also investigate spectral synthesis and analysis in the class $C(G,E)$. It is known that if $G$ is a compact Abelian group and $E$ is a Banach space, then spectral synthesis holds in $C(G,E)$. On the other hand, we show that if $G$ is an infinite and discrete Abelian group and $E$ is a Banach space of infinite dimension, then even spectral analysis fails in $C(G,E)$. If, however, $G$ is discrete, has finite torsion free rank and if $E$ is a Banach space of finite dimension, then spectral synthesis holds in $C(G,E)$.
Let $G$ be a topological commutative semigroup with unit. We prove that a continuous function $fcolon Gto cc$ is a generalized exponential polynomial if and only if there is an $nge 2$ such that $f(x_1 +ldots +x_n )$ is decomposable; that is, if $f(x
_1 +ldots +x_n )=sumik u_i cd v_i$, where the function $u_i$ only depends on the variables belonging to a set $emp e E_i subsetneq { x_1 stb x_n }$, and $v_i$ only depends on the variables belonging to ${ x_1 stb x_n } se E_i$ $(i=1stb k)$.
