Fyrirlesari: Benedikt Steinar Magnússon (Háskóli Íslands)
Titill: Liouville’s theorem: \(L^2\) variant for graded polynomial rings
Staðsetning: 1 desember 2023, kl. 11:40 í stofu 152 í VR-II
Ágrip: Liouville’s theorem is a fundamental theorem in complex analysis which states that every bounded entire function must be constant. A corollary of it shows that an entire function which grows like a polynomial must be a polynomial. The original Liouville theorem uses the sup-norm to determine growth.
We will present a result which states that an entire function satisfying a weighted \(L^2\) estimate is a polynomial. Recently there has been some development in studying the connections between potential theory in several complex variables and graded polynomial rings. Then the standard grading of polynomials (which determines its degree) is replaced with a grading depending on a compact convex set S in \(\mathbb R_+^n\).
The \(L^2\)-weight in our result depends on S and the corresponding polynomial will have its exponents in a dilate of a certain hull of S, which determines its „degree“. Furthermore, we will show that in some nice cases this hull equals S and an example which shows that in general the hull must be strictly larger than S.