Algebraic versions of Hartogs' theorem (Q6608791)

The paper finds families \(\mathcal A\) of test curves such that a given function \(f:\mathbb K^n \to \mathbb K\) is regular if and only if the restriction \(f|_C\) of \(f\) to every curve \(C\) in \(\mathcal A\) are regular, where \(\mathbb K\) is a field and \(n \geq 2\).\N\NThe first main theorem says that we can choose the family of nonsingular curves as \(\mathcal A\) if and only if \(\mathbb K\) is uncountable and algebraically closed. Its proof is constructive and non-regular functions whose restriction to nonsingular algebraic curves are regular are constructed when \(\mathbb K\) does not satisfy the condition.\N\NThe first main theorem implies that \(\mathcal A\) must contains a singular algebraic curve when \(\mathbb K\) does not satisfy the above condition. The paper considers the case where \(\mathbb K\) is uncountable and of characteristic zero. The second main theorem says that, in such a case, we can choose the set of affine lines parallel to coordinate axes and translates of curves in an `admissible' family as \(\mathcal A\) when \(n=2\). When \(n>2\), the family of the images of such curves under affine isomorphisms of \(\mathbb K^2\) onto \(2\)-planes in \(\mathbb K^n\) is sufficient. An admissible family is defined as a collection \(\mathcal E\) of algebraic curves in \(\mathbb K^2\) passing the origin such that, for every integer \(l\), the multiplicity of the defining polynomial of some curve in \(\mathcal E\) at the origin is larger than \(l\). Two examples of admissible families are given, and one is the family of curves defined by the equation \(X^p-Y^q=0\), where \(p<q\) are prime numbers.