Strong exponent bounds for the local Rankin-Selberg convolution

Let $F$ be a non-Archimedean locally compact field. Let $\sigma$ and $\tau$ be finite-dimensional semisimple representations of the Weil-Deligne group of $F$. We give strong upper and lower bounds for the Artin and Swan exponents of $\sigma\otimes\tau$ in terms of those of $\sigma$ and $\tau$. We give a different lower bound in terms of $\sigma\otimes\check\sigma$ and $\tau\otimes\check\tau$. Using the Langlands correspondence, we obtain the bounds for Rankin-Selberg exponents.