Complexity of membership problems of different types of polynomial ideals

We survey degree bounds and complexity classes of the word problem for polynomial ideals and related problems. The word problem for general polynomial ideals is known to be exponential space-complete, but there are several interesting subclasses of polynomial ideals that allow for better bounds.We review complexity results for polynomial ideals with low degree, toric ideals, binomial ideals, and radical ideals. Previously known results as well as recent findings in our project “Degree Bounds for Gröbner Bases of Important Classes of Polynomial Ideals and Efficient Algorithms” are presented.