This paper proves equationality of the theories $\"SCF$ of separably closed fields of arbitrary inseparability degree, and of the theories $\"ACFP$ 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 $\"ACFP_0$ by a result of Günaydın [5]. Equationality means that every formula is equivalent to a boolean combination of equations. In the case of $\"SCF$, the sufficient set of equations they find (``$\lambda$-tame formulas'') is defined recursively in terms of the $\lambda$-functions. For $\"ACFP$, 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 $\"DCF_p$ and as a consequence an alternative proof of equationality of $\"SCF_{p,\infty}$, is available on the arXiv.

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