A semialgebraic set is defined by finite unions and complements of inequalities of the form $g(x)\ge 0$ where $g$ is a multivariate polynomial with integer coefficients.
My question considers the extension of this to the case where $g$ might have irrational exponents. For example, the set $\{(x,x^{\log_2 5}): x\ge 1\}$ is not semialgebraic.
First, does this model have a common name? Was it studied anywhere?
Clearly some of the nice computational properties of semialgebraic sets are lost, since it's not even clear how to represent the exponents effectively. However, many properties should be retained, such as o-minimality and CAD.
Anyone knows any useful references for this?