부산대학교 도서관

Suppose that $f(x_1,\dots,x_n)$ is a homogeneous form of degree $d$with coefficients in a field $K\subseteq \Bbb{C}$. Consider the {\it $K$-length} of $f$, which is defined to be the smallest $r$ for whichthere exist $\lambda_j$, $\alpha_{jk}\in K$ such that$$f(x_1,\dots,x_n)=\sum_{j=1}^r\lambda_j(\alpha_{j1}x_1+\cdots+\alpha_{jn}x_n)^d.$$The $K$-length is an important subject in the algebraic invarianttheory and has been studied extensively in the literature. The paperunder review primarily deals with the binary case, namely $n=2$. Manyold and new results are proved in this paper. In particular, the authorgives new elementary proofs of two classical theorems of Sylvesterwhich were first established in 1851 and 1864 respectively. Then someinteresting applications of the method are discussed. For instance, fora fixed form $f$, how the $K$-length of $f$ varies as the field $K$varies is a very interesting and difficult question. The author alsodiscusses some related open questions at the end of the paper.