A coincidence theorem for symmetric multilinear forms
Let E be a vector space over an algebraically closed field K, \(D(K_{\omega})\) be a generalized circular region (in the case \(K={\mathbb{C}}^ a \)subset of \({\mathbb{C}}_{\omega}={\mathbb{C}}\cup \infty\) which is bounded by a circle or straight line and which contains a connected subset (possibly empty) of their boundary). For a given \(N\subseteq E^ 2\) and a mapping G: \(N\to D(K_{\omega})\) the circular cone \(E_ 0(N;G)\) in E is defined by \(E_ 0(N;G)=\cup T_ G(x,y)\), where \(T_ G(x,y)=\{sx+ty\neq 0| s,t\in K\); s/t\(\in G(x,y)\}\) and the union ranges over all (x,y)\(\in N\). The main result of the paper generalizes the Walsh coincidence theorem to symmetric multilinear forms: Theorem 3.3. Let \(P(x_ 1,x_ 2,...,x_ n)\) be a symmetric n-linear form from \(E^ n\) to K and \(E_ 0(N;G)\) be a circular cone in E. If \(\xi_ j\in T_ G(x_ 0,y_ 0)\), \(j=1,...,n\), for some \((x_ 0,y_ 0)\in N\) such that \(P(\xi_ 1,\xi_ 2,...,\xi_ n)=0\) then there exists an element \(\xi \in T_ G(x_ 0,y_ 0)\) such that \(P(\xi,\xi,...,\xi)=P(\xi_ 1,\xi_ 2,...,\xi_ n)\). As an application it is shown how this theorem leads to simple alternative proofs of a number of known results. Also a vector-valued version of the theorem is given.
scientific article; zbMATH DE number 4069102
scientific article