A regular interpolation problem and its applications

Abstract

In this article, we prove the following interpolation problem: if the composition of a function and a regular map between affine varieties is a regular function, then there exists a global regular function of the target variety that coincide with the function on the image of the regular map provided the target variety is factorial and the regular map is almost surjective. We also discuss a few applications of the interpolation problem.

Communicated by Daniel Erman

Keywords:

• Affine variety
• almost surjective morphism
• analytic space
• factorial variety

2020 Mathematics Subject Classification:

• 14M05
• 14R10
• 13B25
• 32C15

Acknowledgments

The author would like to thank Chitrabhanu Chaudhuri, Ritwik Mukherjee, and Zbigniew Jelonek for several fruitful discussions and comments. Conversations with Erhard Aichinger over email were helpful. Special thanks are due to him for pointing out reference [1], which the author had looked at for a different purpose before beginning the project. The author is grateful to Chitrabhanu Chaudhuri for providing an example to sort out the mistake that the author had made earlier. The author is also thankful to the anonymous referee for suggesting several improvements to the earlier manuscript.

Notes

1 For $f,g\in K[X]$, if $r={\text{Res}}_X(f,g)$, then r = 0 if and only if both f and g have a non-constant common factor in $K[X]$ (cf. [3]).

Additional information

Funding

This work is supported by the INSPIRE faculty fellowship (Registration No.: IFA21-MA 161) funded by the DST, Govt. of India.