Further study of $2$-to-$1$ mappings over $\mathbb{F}_{2^n}$

$2$-to-$1$ mappings over finite fields play an important role in symmetric cryptography, in particular in the constructions of APN functions, bent functions, semi-bent functions and so on. Very recently, Mesnager and Qu \cite{MQ2019} provided a systematic study of $2$-to-$1$ mappings over finite fields. In particular, they determined all $2$-to-$1$ mappings of degree at most 4 over any finite fields. In addition, another research direction is to consider $2$-to-$1$ polynomials with few terms. Some results about $2$-to-$1$ monomials and binomials have been obtained in \cite{MQ2019}. Motivated by their work, in this present paper, we push further the study of $2$-to-$1$ mappings, particularly, over finite fields with characteristic $2$ (binary case being the most interesting for applications). Firstly, we completely determine $2$-to-$1$ polynomials with degree $5$ over $\mathbb{F}_{2^n}$ using the well known Hasse-Weil bound. Besides, we consider $2$-to-$1$ mappings with few terms, mainly trinomials and quadrinomials. Using the multivariate method and the resultant of two polynomials, we present two classes of $2$-to-$1$ trinomials, which explain all the examples of $2$-to-$1$ trinomials of the form $x^k+\beta x^{\ell} + \alpha x\in\mathbb{F}_{{2^n}}[x]$ over $\mathbb{F}_{{2^n}}$ with $n\le 7$, and derive twelve classes of $2$-to-$1$ quadrinomials with trivial coefficients over $\mathbb{F}_{2^n}$.