Two Absolutely Irreducible Polynomials over ${\mathbb{F}}_2$ and Their Applications to a Conjecture by Carlet

Two polynomials $F_k(X_1,\dots ,X_k)$ and ${\mathrm{\Theta }}_k(X_1,\dots ,X_k)$ over ${\mathbb{F}}_2$ arose from the study of a conjecture by C. Carlet about the sum-freedom of the multiplicative inverse function of ${\mathbb{F}}_{2^n}$. Both $F_k$ and ${\mathrm{\Theta }}_k$ are homogeneous and symmetric with $\text{deg}{F}_k=2^k-2$ and $\text{deg}{\mathrm{\Theta }}_k=2^{k-1}$. It is known that $F_k$ is absolutely irreducible for $k\ge 3$. Using the Lang-Weil bound and a curious connection between $F_k$ and ${\mathrm{\Theta }}_k$, we show that ${\mathrm{\Theta }}_k$ ($k\ge 3$) is also absolutely irreducible. This conclusion allows us to improve several existing results about Carlet’s conjecture.