matrices that has been studied in [@Ando98], [@Pablo_Jadbabaie_JSR_journal] mainly because it provides a “worst-case” example for the method of common quadratic Lyapunov functions. Indeed, it is easy to show that $\rho(\mathcal{A})=1$, but a common quadratic Lyapunov function can only produce an upper bound of $\sqrt{2}$ on the JSR. In [@Pablo_Jadbabaie_JSR_journal], Parrilo and Jadbabaie give a simple degree-4 (non-convex) common polynomial Lyapunov function that proves stability of the switched linear system defined by the matices $\{\gamma A_1, \gamma A_2\},$ for any $\gamma<1$. In sharp contrast, we show the following:

\[thm:degree.higher\] Let $A_1,A_2$ be as in (\[eq:A1,A2.ando.shi\]) and consider the sets of matrices ${{\cal{M}}}_\gamma=\{\gamma A_1, \gamma A_2\}$ parameterized by a scalar $\gamma <1$. As $\gamma\rightarrow 1$, the minimum degree of a common convex polynomial Lyapunov function for ${{\cal{M}}}_\gamma$ goes to infinity.

It is sufficient to prove that the set $\{A_1,A_2\}$ has no convex invariant set defined as the sublevel set of a polynomial. Indeed, if there were a uniform bound $D$ on the degree of a convex polynomial Lyapunov function this would imply the existence of an invariant set—which is the sublevel set of a convex polynomial function of degree $D$—\
We prove our claim by contradiction. In fact, we will prove 