A ternary algebra with applications to binary quadratic forms (Q2765430)

Let \(q(x,y)= ax^2+ bxy+ cy^2+ dx+ ey+ f\in \mathbb{Z} [x,y]\) be a general binary quadratic polynomial with \(\text{Dis} (q)= ac- \frac{b^2}{4}\) and \(\text{Det} (q)=\) determinant of the \(3\times 3\) symmetric matrix attached to \(q\). For fixed \(q\) with \(\text{Dis} (q)\neq 0\) the author defines a \(\mathbb{Q}\)-vector space \(A(q)\) and a commutative ring \(R(q)\) consisting of certain rational \(2\times 2\) matrices. He proves that \(A(q)\) is a left \(R(q)\)-module. Furthermore the map \(A(q)\times A(q)\to R(q)\) given by \((A,B)\mapsto A\cdot B^*\), where \(B^*\) is the transpose of the cofactor matrix of \(B\), is surjective. NEWLINENEWLINENEWLINEThe triple product \((A,B,C)\mapsto A\cdot B^*\cdot C\) on \(A(q)\) satisfies associative, distributive and linearity conditions such that \(A(q)\) is a ``ternary algebra'' over \(\mathbb{Q}\). (Note that \(A(q)\) is not closed under multiplication unless \(a=1\)). NEWLINENEWLINENEWLINEIf \(\text{Det} (q)\cdot \text{Dis} (q)\neq 0\) and if the curve \(C(q)= \{(x,y)\in \mathbb{Q}: q(x,y)= 0\}\) is nonempty then \(C(q)\) is a ``commutative ternary group'', in particular we have \(P\cdot Q^*\cdot R\in C(q)\) for \(P,Q,R\) in \(C(q)\) under the natural map \(C(q)\hookrightarrow A(q)\). NEWLINENEWLINENEWLINEThere are two nice applications: NEWLINENEWLINENEWLINE1. If \(q\) is a (homogeneous) quadratic form, i.e. if \(d= e= f= 0\), then \(q(x_1,y_1) q(x_2,y_2) q(x_3,y_3)= q(x,y)\) where NEWLINE\[NEWLINEx= a(x_1x_2x_3)+ b(x_1y_2x_3)+ c(x_1y_2y_3- y_1x_2y_3+ y_1y_2x_3)NEWLINE\]NEWLINE and a similar trilinear expression for \(y\). NEWLINENEWLINENEWLINE2. If \(\mathbb{Q}(q)= \{\alpha= q(x,y)\mid x,y\in \mathbb{Q}\), \(\alpha\operatorname {Dis}(q)\neq \text{Det}(q)\}\) is nonempty then it is a commutative ternary group under the map NEWLINE\[NEWLINE(\alpha,\beta,\gamma)\mapsto \alpha \beta^* \gamma= \frac {\alpha \gamma \operatorname {Dis} (q)- (\alpha- \beta+ \gamma) \text{Det} (q)} {\beta \operatorname {Dis} (q)- \text{Det} (q)}.NEWLINE\]NEWLINE These formulas generalize well-known classical formulas for multiplicative quadratic forms. Apart from a few misprints the paper is very well written.NEWLINENEWLINEFor the entire collection see [Zbl 0972.00026].