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 which there 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 invariant theory and has been studied extensively in the literature. The paper under review primarily deals with the binary case, namely $n=2$. Many old and new results are proved in this paper. In particular, the author gives new elementary proofs of two classical theorems of Sylvester which were first established in 1851 and 1864 respectively. Then some interesting applications of the method are discussed. For instance, for a fixed form $f$, how the $K$-length of $f$ varies as the field $K$ varies is a very interesting and difficult question. The author also discusses some related open questions at the end of the paper.