论文标题
矩阵形态和高度形式的组成，并应用于二芬太汀方程
Matrix morphology and composition of higher degree forms with applications to diophantine equations
论文作者
论文摘要
In this paper we use matrices, whose entries satisfy certain linear conditions, to obtain composition identities $f(x_i)f(y_i)=f(z_i)$, where $f(x_i)$ is an irreducible form, with integer coefficients, of degree $n$ in $n$ variables ($n$ being $3,\,4,\,6$ or $8$), and $ x_i，\，y_i，\; i = 1，\，\，\，\，\ ldots，\，n $是自变量，而$ z_i，\; i = 1，\; i = 1，\，\，2，\，\ ldots，\ ldots，\，\，n $的值是由birineAr form bilinear forminear the birineAr形式给出的。当$ n = 2，\，4 $或$ 8 $时，我们还获得了构图身份$ f（x_i）f（y_i）f（y_i）f（z_i）= f（w_i）$在哪里，与以前一样，$ f（x_i）$是一种不可修复的形式，具有整数系数 $x_i,\,y_i,z_i,\;i=1,\,2,\,\ldots,\,n$, are independent variables and the values of $w_i,\;i=1,\,2,\,\ldots,\,n$, are given by trilinear forms in the variables $x_i,\,y_i,\,z_i$, and因此，身份不能从类型$ f（x_i）f（y_i）= f（z_i）$的任何身份得出。此外，我们描述了一种获得较高程度形式的这两种类型的组成身份的方法。本文中给出的构图身份尚未获得。我们还可以在某些四分之一和八个二只方程式的积极整数中获得无限的解决方案$ f（x_1，\，\，\ ldots，\，x_n）= 1 $ whene $ f（x_1，\，\ ldots，\ ldots，\，\，x_n）$是一种允许组成身份的形式，可以允许构成身份和$ n = 4 $或$ 8 $ 8 $ 8 $ 8 $ 8 $ 8 $ 8 $ 8 $或8 $ 8 $ 8 $ 8 $ 8 $ 8 $ 8 $。
In this paper we use matrices, whose entries satisfy certain linear conditions, to obtain composition identities $f(x_i)f(y_i)=f(z_i)$, where $f(x_i)$ is an irreducible form, with integer coefficients, of degree $n$ in $n$ variables ($n$ being $3,\,4,\,6$ or $8$), and $x_i,\,y_i,\;i=1,\,2,\,\ldots,\,n$, are independent variables while the values of $z_i,\;i=1,\,2,\,\ldots,\,n$, are given by bilinear forms in the variables $x_i,\,y_i$. When $n=2,\,4$ or $8$, we also obtain composition identities $f(x_i)f(y_i)f(z_i)=f(w_i)$ where, as before, $f(x_i)$ is an irreducible form, with integer coefficients, of degree $n$ in $n$ variables while $x_i,\,y_i,z_i,\;i=1,\,2,\,\ldots,\,n$, are independent variables and the values of $w_i,\;i=1,\,2,\,\ldots,\,n$, are given by trilinear forms in the variables $x_i,\,y_i,\,z_i$, and such that the identities cannot be derived from any identities of the type $f(x_i)f(y_i)=f(z_i)$. Further, we describe a method of obtaining both these types of composition identities for forms of higher degrees. The composition identities given in this paper have not been obtained earlier. We also obtain infinitely many solutions in positive integers of certain quartic and octic diophantine equations $f(x_1,\,\ldots,\,x_n)=1$ where $f(x_1,\,\ldots,\,x_n)$ is a form that admits a composition identity and $n=4$ or $8$.
