Problems / algebra

algebra / Mathieu–Zhao Spaces, Integral Conjectures

Counterexamples to the xz-Conjecture and the Mathieu Conjecture for SU(2)

For the integral $\mathcal{I}(h)$ over the unit interval and the torus, the paper gives the three-term Laurent polynomial $f(x,z)=(1-z^{-1})((1-x)+xz)$ with $\mathcal{I}(f^n)=0$ but $\mathcal{I}(z^{-1}f^n)=(-1)^{n-1}/(n+1)\neq0$. This disproves the $xz$-conjecture with one interval and one torus variable, shows $\ker\mathcal{I}$ is not a Mathieu–Zhao subspace, and by padding yields counterexamples for SU(2).