Why is the computation of the general $p$-norm hard?

Question

Consider a $2 \times 2$ matrix. Can we find a general polynomial for computing the operator norm induced by a $p$-norm?

Answer 1

The norm of the matrix $\pmatrix{a & b\cr c & d\cr}$ is the maximum of $(|a x + b y|^p + |c x + d y|^p)^{1/p}$ subject to the constraint $|x|^p + |y|^p = 1$. Using a Lagrange multiplier, we would consider $$F(x,y,\lambda) = |ax+by|^p + |cx+dy|^p + \lambda(|x|^p + |y|^p - 1)$$ take the derivatives with respect to $x$, $y$ and $\lambda$, and solve. If $p$ is rational, the results will involve algebraic functions of $a,b,c,d$, but are likely to be very complicated. For example, in the case $p=4$, $x$ will be a root of a polynomial of degree $24$ with coefficients polynomials in $a,b,c,d$, with $340$ terms when expanded. If $p$ is irrational, I don't see how they could be algebraic functions in general.