Certificate Complexity of Elementary Symmetric Boolean Functions of Arbitrary Degree

Let $σ_{n,d}$ denote the elementary symmetric Boolean function of $n$ variables and degree $d$. Previous work determined $C(σ_{n,d})$ when $d$ is odd or a power of two, but the general even non-power-of-two case remained open. We determine the certificate complexity for every degree $1\le d\le n$, t