Delta-rational chord: Difference between revisions
m →Facts: I haven't proven this yet |
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<math>\displaystyle{r_{\mathbf{u}, \mathbf{v}}(x) = \frac{2^{v_p}x^{v_g}- 2^{u_p}x^{u_g}}{2^{u_p}x^{u_g} - 1} }.</math> | <math>\displaystyle{r_{\mathbf{u}, \mathbf{v}}(x) = \frac{2^{v_p}x^{v_g}- 2^{u_p}x^{u_g}}{2^{u_p}x^{u_g} - 1} }.</math> | ||
Then for any positive rational number <math>m/n</math> in the image <math>r_{\mathbf{u}, \mathbf{v}}(I),</math> there exists <math>g \in I</math> that satisfies <math>r_{\mathbf{u}, \mathbf{v}}(g) = m/n,</math> making the specified chord ('''0''', '''u''', '''v''') a +'' | Then for any positive rational number <math>m/n</math> in the image <math>r_{\mathbf{u}, \mathbf{v}}(I),</math> there exists <math>g \in I</math> that satisfies <math>r_{\mathbf{u}, \mathbf{v}}(g) = m/n,</math> making the specified chord ('''0''', '''u''', '''v''') a +''n''+''m'' DR chord. | ||
(TODO: Case analysis according to whether <math>r_{\mathbf{u}, \mathbf{v}}(x)</math> simplifies to a polynomial.) | (TODO: Case analysis according to whether <math>r_{\mathbf{u}, \mathbf{v}}(x)</math> simplifies to a polynomial.) | ||