Polynomially presented properties: preservation proposition perspectives

Abstract: Many properties of a field (or of some of its elements) can be captured by asking whether certain polynomial equations have solutions in the field. Examples include statements like “$x$ can be written as a sum of two cubes” ($x = y^3 + z^3$) or “some non-zero element is simultaneously a square and minus a square’’ ($zy^2 = 1 = -zw^2$). When given such a property, what is its most efficient description via polynomial equations, that is, with the lowest number of variables?
Model theory offers a helpful perspective through results known as preservation theorems. These (meta)theorems reflect a simple guiding idea: if a property behaves like it comes from a certain kind of description, then it actually does. To use a known trope: if it looks, swims, and quacks like a duck, it is a duck.
I will introduce some famous preservation theorems, and a less famous one which presents a way to think about the problem from the first paragraph. If there is time, I will also touch on a more geometric viewpoint on this problem, and some surprisingly simple related open questions.
This talk is partially based on joint work with Arno Fehm and Philip Dittmann.

This talk was given at the Algebra Colloquium of Charles University’s Faculty of Mathematics and Physics. You may find the slides for this talk here.