sum $$p^k(x)-\left(p(A_jx)+\alpha_j (\sum_i x_i^2)^d\right)^k$$ itself is sos. Now note that we can write $$\begin{aligned}
    &p^k(x)-\left(p(A_jx)+\alpha_j (\sum_i x_i^2)^d\right)^k=p(x)^k-p(A_j)^k -\sum_{s=1}^k \binom{k}{s} \alpha_j^s (\sum_i x_i^2)^{ds}\cdot p(A_jx)^{k-s},
    \end{aligned}$$ which enables us to conclude that $$q(x)-q(A_jx)=p^k(x)-\left(p(A_jx)+\alpha_j (\sum_i x_i^2)^d\right)^k+\sum_{s=1}^k \binom{k}{s} \alpha_j^s (\sum_i x_i^2)^{ds}\cdot p(A_jx)^{k-s}$$ is sos as the sum of sos polynomials.

Non-existence of a uniform bound on the degree of convex polynomial Lyapunov functions
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It is known that there are families of $n\times n$ matrices ${{\cal{M}}}=\{A_1,\ldots,A_m\}$ for which the switched linear system (\[eq:switched.linear.system\]) is asymptotically stable under arbitrary switching, but such that the minimum degree of a common polynomial Lyapunov function is arbitrarily large [@aaa_raph_lower_bounds]. (In fact, this is the case already when $m=n=2$.) In the case where ${{\cal{M}}}$ admits a common polynomial Lyapunov function of degree $d$, it is natural to ask whether one can expect ${{\cal{M}}}$ to also admit a common *convex* polynomial Lyapunov function of some degree $\hat{d}$, where $\hat{d}$ is a function of $d,n,m$ only? In this subsection, we answer this question in the negative.

Consider the set of matrices $\mathcal{A}=\{A_1,A_2\},$ with $$\label{eq:A1,A2.ando.shi}
A_{1}=\left[
\begin{array}
[c]{cc}1 & 0\\
1 & 0
\end{array}
\right]  ,\text{ }A_{2}=\left[
\begin{array}
[c]{cr}0 & 1\\
0 & -1
\end{array}
\right].$$ This is a benchmark set of 