q2 — homogeneous plane Keller maps

This route started from the imagined end-state “classify every homogeneous plane perturbation.” It closes, but only by recovering a classical shear family, so it does not improve the degree record.

Let $H=(R,S)$ be homogeneous of degree $d\ge2$ , and suppose $F=I+H$ has Jacobian determinant 1. In

\det JF=1+(R_x+S_y)+(R_xS_y-R_yS_x),

the two nonconstant terms have distinct degrees. Thus the divergence and $\det JH$ vanish separately. The divergence identity gives

R=h_y,\qquad S=-h_x

for a binary homogeneous form $h$ of degree $d+1$ , while the second identity says that the Hessian determinant of $h$ is zero.

For completeness, dehomogenize as $h=x^m f(y/x)$ . Its Hessian determinant, apart from the nonzero factor $(m-1)x^{2m-4}$ , is

mff''-(m-1)(f')^2.

At a root of multiplicity $r$ , vanishing forces $r=m$ . Hence $f$ is constant or has one root of multiplicity $m$ , so $h$ is a power of a linear form. After absorbing constants, every map in the class has the form

F(x,y)=\bigl(x+cb(ax+by)^d,\ y-ca(ax+by)^d\bigr).

The linear form $ax+by$ is invariant, so changing the two nonlinear signs gives the polynomial inverse. verify_family.py checks the Jacobian, invariant, and both compositions in exact sparse integer arithmetic, including degree 125.

./problems/smale-jacobian-n2/compute/q2/run_all.sh

Final marker: Q2_HOMOGENEOUS_CLASS_PASS.