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algcurves

 is_hyperelliptic
 Test if an algebraic curve is hyperelliptic

 Calling Sequence is_hyperelliptic(f, x, y)

Parameters

 f - irreducible polynomial in x and y x - variable y - variable

Description

 • An irreducible algebraic curve $f\left(x,y\right)$ is called rational if the genus is 0. In this case a parametrization can be computed. It is called elliptic if the genus is 1. It is called hyperelliptic if the genus is more than 1 and $f\left(x,y\right)$ is birational to a curve $F\left(X,Y\right)$ with degree(F,Y)=2. In the elliptic or hyperelliptic case a normal form can be computed with Weierstrassform.
 • If the genus is less than 2, then the curve is not hyperelliptic and the output will be false. If the genus is 2, then the output will be true. For higher genus, the output can be true or false.

Examples

 > $\mathrm{with}\left(\mathrm{algcurves}\right):$
 > $f≔{y}^{2}+{x}^{5}+1$
 ${f}{≔}{{x}}^{{5}}{+}{{y}}^{{2}}{+}{1}$ (1)
 > $g≔{y}^{9}+2{x}^{2}{y}^{6}+2{x}^{4}{y}^{3}+{x}^{6}+{y}^{2}$
 ${g}{≔}{{y}}^{{9}}{+}{2}{}{{x}}^{{2}}{}{{y}}^{{6}}{+}{2}{}{{x}}^{{4}}{}{{y}}^{{3}}{+}{{x}}^{{6}}{+}{{y}}^{{2}}$ (2)
 > $h≔{y}^{9}+3{x}^{2}{y}^{6}+3{x}^{4}{y}^{3}+{x}^{6}+{y}^{2}$
 ${h}{≔}{{y}}^{{9}}{+}{3}{}{{x}}^{{2}}{}{{y}}^{{6}}{+}{3}{}{{x}}^{{4}}{}{{y}}^{{3}}{+}{{x}}^{{6}}{+}{{y}}^{{2}}$ (3)
 > $\mathrm{is_hyperelliptic}\left(f,x,y\right)$
 ${\mathrm{true}}$ (4)
 > $\mathrm{is_hyperelliptic}\left(g,x,y\right)$
 ${\mathrm{false}}$ (5)
 > $\mathrm{is_hyperelliptic}\left(h,x,y\right)$
 ${\mathrm{true}}$ (6)
 > $\mathrm{Weierstrassform}\left(h,x,y,\mathrm{x0},\mathrm{y0}\right)$
 $\left[{-}{{\mathrm{x0}}}^{{7}}{+}{7}{}{{\mathrm{x0}}}^{{6}}{-}{21}{}{{\mathrm{x0}}}^{{5}}{+}{35}{}{{\mathrm{x0}}}^{{4}}{-}{35}{}{{\mathrm{x0}}}^{{3}}{+}{21}{}{{\mathrm{x0}}}^{{2}}{+}{{\mathrm{y0}}}^{{2}}{-}{7}{}{\mathrm{x0}}{+}{2}{,}\frac{{y}{}\left({{y}}^{{3}}{+}{{x}}^{{2}}{+}{1}\right)}{{{y}}^{{6}}{+}{2}{}{{x}}^{{2}}{}{{y}}^{{3}}{+}{{x}}^{{4}}{+}{y}}{,}{-}\frac{{x}{}\left({{y}}^{{3}}{+}{{x}}^{{2}}\right)}{{y}}{,}{-}\left({\mathrm{x0}}{-}{1}\right){}{\mathrm{y0}}{,}{-}{{\mathrm{x0}}}^{{3}}{+}{3}{}{{\mathrm{x0}}}^{{2}}{-}{3}{}{\mathrm{x0}}{+}{1}\right]$ (7)

 See Also