Cohomology of Lie algebras of polynomial vector fields on the line over fields of characteristic $2$

For a field $\mathbb{F}$, let $L_k(\mathbb{F})$ be the Lie algebra of derivations $f(t)\frac{d}{dt}$ of the polynomial ring $\mathbb{F}[t]$, where $f(t)$ is a polynomial of degree $\geqslant k$. For any $k\geqslant -1$, we present a basis of the space of the cohomology with finite-dimensional support of the Lie algebra $L_k(\mathbb{F})$ with coefficients in the trivial module $\mathbb{F}$ for the case where ${\rm char}(\mathbb{F})=2$. The main result obtained is an analog of the famous Goncharova's Theorem for the case ${\rm char}(\mathbb{F})=0$ and $k\geqslant 1$.