12 Quadratic relations near a point of a circle
Results of the library’s work on Diaz’s conjecture about quadratic relations at a point \(u\) of a circle of algebraic radius, that is \(u\neq 0\) with \(\rho =u\bar u\) algebraic. On such a point the norm form \(X_1X_2-\rho X_0^{2}\) vanishes at \((1,u,\bar u)\). Theorems 12.2 and 12.3 show that on generic data it is the only quadratic relation, and that the period \(i\pi \) never enters a singular \(2\times 2\) matrix; Theorems 12.5 and 12.6 describe the matrix a counterexample to Diaz’s conjecture would give, and show that its only singular pencil is the norm form. None of the four statements involves \(e^{u}\), and none was found in the sources read (Chapter 14).
12.1 Matrices of linear forms
Let \(F\) be a field of characteristic \(0\) and let \(M\) be a \(2\times 2\) matrix whose entries are linear forms over \(F\) in \(n\) variables. If \(\det M\) vanishes identically, then the rows of \(M\) are linearly dependent over \(F\), or its columns are.
Source: Elementary.
Write \(M=\begin{pmatrix} L_{11} & L_{12} \\ L_{21} & L_{22} \end{pmatrix}\), so that \(L_{11}L_{22}=L_{12}L_{21}\) in \(F[X_1,\dots ,X_n]\). If \(L_{11}=0\), then \(L_{12}=0\) or \(L_{21}=0\), a zero row or a zero column. Otherwise the linear form \(L_{11}\) is irreducible and divides \(L_{12}L_{21}\), so \(L_{12}=cL_{11}\) or \(L_{21}=cL_{11}\) with \(c\in F\); cancelling \(L_{11}\) gives \(L_{22}=cL_{21}\), a column relation, or \(L_{22}=cL_{12}\), a row relation.
12.2 Generic data
Let \(u\neq 0\) with \(\rho =u\bar u\) algebraic, and suppose that \(u\) and \(i\pi \) are algebraically independent over \(\overline{\mathbb {Q}}\). Every quadratic form with algebraic coefficients that vanishes at \((1,u,\bar u,i\pi )\) is an algebraic multiple of \(X_1X_2-\rho X_0^{2}\).
Source: Not found in the sources read, and not routine (Chapter 14). No source read considers the triple \((u,\bar u,i\pi )\).
Substitute \(\bar u=\rho /u\) and multiply by \(u^{2}\). The form becomes a polynomial in \(u\) and \(i\pi \) with nine monomials, and by independence all its coefficients vanish. The only two terms of the form that land on the same monomial are \(X_0^{2}\) and \(X_1X_2\), both at \(u^{2}\); their coefficients are therefore proportional to \(-\rho \) and \(1\), and every other coefficient is zero.
Let \(u\neq 0\) with \(\rho =u\bar u\) algebraic, and suppose that \(u\) and \(i\pi \) are algebraically independent over \(\overline{\mathbb {Q}}\). Let \(M\) be a singular \(2\times 2\) matrix with entries \(c_{ij0}+c_{ij1}u+c_{ij2}\bar u+c_{ij3}\, i\pi \), \(c_{ijk}\) algebraic, whose rows are linearly independent over \(\overline{\mathbb {Q}}\) and whose columns are linearly independent over \(\overline{\mathbb {Q}}\). Then \(c_{ij3}=0\) for all \(i,j\): the coefficient of \(i\pi \) vanishes in every entry.
Source: Not found in the sources read, and not routine (Chapter 14). No source read considers the triple \((u,\bar u,i\pi )\).
Write \(M=\sum _k x_kC_k\) at \(x=(1,u,\bar u,i\pi )\) with constant matrices \(C_k\). By Theorem 12.2, \(\det \bigl(\sum _k x_kC_k\bigr)=c\, (x_1x_2-\rho x_0^{2})\) identically, and \(c\neq 0\): otherwise Lemma 12.1 over \(\overline{\mathbb {Q}}\) gives dependent rows or columns. For the polar form of the determinant, \(C_3\) is then isotropic and orthogonal to \(C_0,C_1,C_2\), which span a non-degenerate three-dimensional subspace of the four-dimensional space of \(2\times 2\) matrices. Its orthogonal complement is a non-degenerate line, whose only isotropic vector is zero.
12.3 The shape of a counterexample
Let \(K\subseteq \mathbb {C}\) be a subfield and \(x,y\in \mathbb {C}\) with \(y\neq 0\) and \(x/y\) transcendental over \(K\). If \(\sum _{i=0}^{d}c_ix^{i}y^{d-i}=0\) with all \(c_i\in K\), then all \(c_i\) are zero.
Source: Elementary.
Divide by \(y^{d}\): then \(\sum _i c_i(x/y)^{i}=0\) is a polynomial relation over \(K\) satisfied by \(x/y\), so every coefficient vanishes.
Let \(K\subseteq \mathbb {C}\) be a subfield, \(r\in K\), and \(u\notin K\) with \(u\bar u=r^{2}\), and put \(H=\begin{pmatrix} u & r \\ r & \bar u \end{pmatrix}\). Then \(w^{\mathsf T}Hv\neq 0\) for all non-zero \(w,v\in K^{2}\).
Source: Not found in the sources read, and not routine (Chapter 14). The nearest printed passage is [ Roy95 , § 3.2, p. 65 ] : Roy’s rank method cannot exclude points of \(xy=z^{2}\) with \(\overline{\mathbb {Q}}\)-free coordinates, and \((u,\bar u,r)\) is such a point; see also [ Fis01 , p. 186 ] .
Here \(u\neq 0\), and \(r\neq 0\) since \(u\bar u=r^{2}\). The matrix \(H\) has rank one, and \(w^{\mathsf T}Hv=(w_0+(r/u)w_1)(uv_0+rv_1)\). If \(uv_0+rv_1=0\) with \(v\neq 0\), then \(v_0\neq 0\) and \(u=-rv_1/v_0\in K\); if \(uw_0+rw_1=0\) with \(w\neq 0\), likewise. Either way \(u\in K\), a contradiction.
Let \(K\subseteq \mathbb {C}\) be a subfield, let \(u\neq 0\) be transcendental over \(K\) with \(u\bar u\in K\), and let \(A,B,C\) be \(2\times 2\) matrices over \(K\) with \(\det (A+uB+\bar uC)=0\). Then there is \(c\in K\) with
Source: Not found in the sources read, and not routine (Chapter 14). Roy has neither this classification nor Theorem 12.5; the nearest printed passage is [ Roy95 , § 3.2, p. 65 ] .
The determinant \(Q(x,y,z)=\det (xA+yB+zC)\) is a quadratic form over \(K\). With \(\rho =u\bar u\), the number \(u^{2}Q(1,u,\rho /u)\) is a polynomial of degree at most \(4\) in \(u\) over \(K\); it vanishes, so all its coefficients vanish. They are the coefficients of \(y^{2}\), \(xy\), \(xz\) and \(z^{2}\) in \(Q\) (up to the factors \(\rho \) and \(\rho ^{2}\)) and \(a_{xx}+\rho \, a_{yz}\); so \(Q=a_{yz}(yz-\rho x^{2})\).