An upper bound for valuations of elliptic nets at primes of bad reduction

Let $E\mathrm{/}K$ be an elliptic curve over a field defined by the short Weierstrass equation and let $\mathbf{P}=(P_1,P_2)\in E(K{)}^2\setminus \{(O,O),(\pm P,\pm P)\}$ be a pair of non-torsion points, where $O$ denotes the point at infinity. We give a uniform upper bound for the quantity $\min (\nu ({\mathrm{\Phi }}_{(v_1,v_2)}(\mathbf{P})),2\nu ({\mathrm{\Psi }}_{(v_1,v_2)}(\mathbf{P})))$, where $\nu$ is a normalised discrete valuation associated to primes where $E$ has bad reduction and ${\mathrm{\Psi }}_{(v_1,v_2)}^2(\mathbf{P})$, ${\mathrm{\Phi }}_{(v_1,v_2)}(\mathbf{P})$ are rational functions for describing $\mathbb{Z}$–linear combination of points $[v_1]P_1+[v_2]P_2$ on $E$, higher–dimensional analogues of division polynomials ${\psi }_n^2,{\phi }_n$ known as net polynomials. They are also elliptic nets, which are $n$–dimensional arrays with values in $K$ satisfying the same nonlinear recurrence relation that division polynomials do as well. Hence, they are useful in studying counting problems on elliptic curves.