We present a method for solving polynomial equations over idempotent ω-continuous semirings. The idea is to iterate over the semiring of functions rather than the semiring of interest, and only evaluate when needed. The key operation is substitution. In the initial step, we compute a linear completion of the system of equations that exhaustively inserts the equations into one another. With func...

Noether’s problem then asks whether K(G) is rational (= purely transcendental) over K. Noether’s problem is closely related to the inverse Galois problem. The main results about Noether’s problem for abelian groups can be found in the survey article [Sw]. More recently, Noether’s problem for nonabelian p-groups was investigated in [CHK, CHPK, HK2, Ka1, Ka2]. Let n ≥ 2 be an arbitrary natural nu...

Periodically driven (Floquet) phases are attractive due to their ability host unique physical phenomena with no static counterparts. We propose a general approach in nontrivially devising square-root version of existing Floquet phases, applicable both noninteracting and interacting setting. The resulting systems found yield richer physics that is otherwise absent the original counterparts robus...

A 2-cover is a multiset of subsets of [n] := {1, 2, . . . , n} such that each element of [n] lies in exactly two of the subsets. A 2-cover is called proper if all of the subsets of distinct, and is called restricted if any two of them intersect in at most one element. In this paper we find asymptotic enumerations for the number of line graphs on n-labelled vertices and for 2-covers. We find tha...