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The spaces of quasi-invariant polynomials were introduced by Chalykh andVeselov [Comm. Math. Phys. 126 (1990), 597-611]. Their Hilbert series overfields of characteristic 0 were computed by Feigin and Veselov [Int. Math. Res.Not. 2002 (2002), 521-545]. In this paper, we show some partial results andmake two conjectures on the Hilbert series of these spaces over fields ofpositive characteristic. On the other hand, Braverman, Etingof and Finkelberg[arXiv:1611.10216] introduced the spaces of quasi-invariant polynomials twistedby a monomial. We extend some of their results to the spaces twisted by asmooth function.

Quasi-Invariants in Characteristic $p$ and Twisted Quasi-Invariants