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Abstract: Given two rational maps $\varphi$ and $\psi$ on $\PP^1$ of degree at leasttwo, we study a symmetric, nonnegative-real-valued pairing $<\varphi,\psi>$which is closely related to the canonical height functions $h \varphi$ and$h \psi$ associated to these maps. Our main results show a strong connectionbetween the value of $<\varphi,\psi>$ and the canonical heights of points whichare small with respect to at least one of the two maps $\varphi$ and $\psi$.Several necessary and sufficient conditions are given for the vanishing of$<\varphi,\psi>$. We give an explicit upper bound on the difference between thecanonical height $h \psi$ and the standard height $h \st$ in terms of$<\sigma,\psi>$, where $\sigmax=x^2$ denotes the squaring map. The pairing$<\sigma,\psi>$ is computed or approximated for several families of rationalmaps $\psi$.



Author: Clayton Petsche, Lucien Szpiro, Thomas J. Tucker

Source: https://arxiv.org/



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