Questions inspired by Mikael Passare’s mathematics

Mikael Passare (1959–2011) was a brilliant mathematician. His PhD thesis from 1984 was a breakthrough in the theory of residues in several complex variables. Some time before 1998 he started to work on amoebas and coamoebas. In discussions with him during the last 30 years many questions have emerged—not all of them were resolved at the time of his premature death. The purpose of the paper is to save from oblivion some of the mathematical ideas of Mikael Passare. In the article some of these unanswered questions are presented, always preceded by a discussion leading up to the question. Some of the questions might present challenges to his nine former PhD students, to his many collaborators around the globe—and to anybody interested. Is there an associative algebra of residue currents and principal-value distributions? Is there an interesting non-associative algebra of such currents? Meromorphic extension using two parameters often leads to points of indeterminacy—what is the natural choice at such points? Several questions have bearing on tropical mathematics. Is it possible to build an axiomatic theory for tropical geometry? There are also questions on tropical polynomials as limits of classical polynomials. Can the absolute values of the coefficients of a polynomial be retrieved from its growth function? Some questions are concerned with digital convexity. Finally, there is a question on the constant term in powers of a Laurent polynomial.