Round forms of higher degree
In this paper, the author considers finite-dimensional forms of degree \(d\geq 2\) over fields \(k\) of characteristic \(0\) or \(>d\), in other words homogeneous polynomials of degree \(d\) in \(n<\infty\) variables that are assumed to be nondegenerate, i.e. the symmetric \(d\)-linear map that is naturally associated to such a form is nondegenerate. \(n\) is called the dimension of the form. There are the usual notions of isomorphism, orthogonal sum, and tensor product of forms. A form of degree \(d\) in the variables \(x_i\), \(1\leq i\leq n\), is called diagonal if it is of the shape \(a_1x_1^d+\ldots +a_nx_n^d\), \(a_i\in k\), and one writes \(\langle a_1,\ldots,a_n\rangle\). For a form \(\varphi\) defined on the \(k\)-vector space \(V\) one defines its value set to be \(D_k(\varphi)=\{\varphi(x)\,|\,x\in V\}\cap k^*\) and one calls \(\varphi\) \(s\)-round for a positive integer \(s\) if \(\varphi\cong a^s\varphi\) for all \(a\in D_k(\varphi)\). Round forms are just \(1\)-round forms, i.e. \(D_k(\varphi)\) is contained in the group of similarity factors \(G_k(\varphi)=\{ a\in k^*\,|\,\varphi\cong a\varphi\}\). Round quadratic forms (the case \(d=2\)) play an important role in the theory of quadratic forms and have been studied extensively. Therefore, the present paper only deals with the case \(d>2\). Various basic properties of round forms are investigated and presented. It is shown that \(|G_k(\varphi)/k^{*d}|\leq n\) for \(\varphi\cong\langle a_1,\ldots,a_n\rangle\) (Theorem 1). For those \(s\)-round forms that satisfy that \(D_k(\varphi)/k^{*d}\) is a nontrivial subgroup of \(k^*/k^{*d}\), the author derives an orthogonal decomposition into certain types of indecomposable forms (Theorems 3 and 4). The paper concludes with some examples and some methods how to construct round forms, for example by using norms of field extensions or norms of structurable algebras.
scientific article; zbMATH DE number 6135273
scientific article