Algebraic versions of Hartogs’ theorem

Let $𝕂$ be an uncountable field of characteristic $0$. For a given function $f:{𝕂}^n\to 𝕂$, with $n\ge 2$, we prove that $f$ is regular if and only if the restriction $f{|}_C$ is a regular function for every algebraic curve $C$ in ${𝕂}^n$ which is either an affine line or is isomorphic to a plane curve in ${𝕂}^2$ defined by the equation $X^p-Y^q=0$, where $p<q$ are prime numbers. We also show that regularity of $f$ can be verified on other algebraic curves in ${𝕂}^n$ with desired geometric properties. Furthermore, if the field $𝕂$ is not algebraically closed, we construct a $𝕂$-valued function on ${𝕂}^n$ that is not regular, but all its restrictions to nonsingular algebraic curves in ${𝕂}^n$ are regular functions.