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This paper proves equationality of the theories $\operatorname{SCF}$ of separably closed fields of arbitrary inseparability degree, and of the theories $\operatorname{ACFP}$ of proper (i.e.\ belles) pairs of algebraically closed fields of arbitrary characteristic. This was previously known for separably closed fields of finite inseperability degree by a result of Srour [18], and for $\operatorname{ACFP}_0$ by a result of Günaydın [5]. Equationality means that every formula is equivalent to a boolean combination of equations. In the case of $\operatorname{SCF}$, the sufficient set of equations they find (``$\lambda$-tame formulas'') is defined recursively in terms of the $\lambda$-functions. For $\operatorname{ACFP}$, by adapting techniques of Günaydın to arbitrary characteristic, they show that a sufficient set of equations is given by the formulas expressing linear dependence over the smaller field of tuples of integer polynomials.

An extended version of this paper, containing in particular a proof of equationality of $\operatorname{DCF}_p$ and as a consequence an alternative proof of equationality of $\operatorname{SCF}_{p,\infty}$, is available on the arXiv.

{\small \em Review licensed under CC-BY-SA 4.0 by Martin Bays 2022}
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