A global Nullstellensatz for ideals of Denjoy-Carleman functions
We prove a Nullstellensatz result for global ideals of Denjoy-Carleman functions in both finitely generated and infinitely generated cases.
math.AG↗
arXiv subjects
Andreea Nicoara
Publications and source records attributed to Andreea Nicoara.
Failure of the Weierstrass Preparation Theorem in quasi-analytic Denjoy-Carleman rings
It is shown that Denjoy-Carleman quasi-analytic rings of germs of functions in two or more variables fail to satisfy the Weierstrass Preparation Theorem. The result is proven via a non-extension theorem.
math.CA↗
A Nullstellensatz for Łojasiewicz ideals
For an ideal of smooth functions that is either Łojasiewicz or weakly Łojasiewicz, we give a complete characterization of the ideal of functions vanishing on its variety in terms of the global Łojasiewicz radical and Whitney closure. We also prove that the Łojasiewicz radical of such an ideal is analytic-like in the sense that its saturation equals its Whitney closure. This allows us to recover in a different way Nullstellensatz results due to Bochnak and Adkins-Leahy and answer positively a modification of the Nullstellensatz conjecture due to Bochnak.
math.AG↗
Global Regularity for the Tangential Cauchy-Riemann Operator on Weakly Pseudoconvex CR Manifolds
Let M of real dimension 2n-1 be a compact, orientable, weakly pseudoconvex manifold of dimension at least five, embedded in C^N (n less than or equal to N), of codimension one or more in C^N, and endowed with the induced CR structure. We show the tangential Cauchy-Riemann operator has closed range on such a manifold M, hence we get global existence and regularity results for the \bar\partial_b problem. We also show the middle (i.e. corresponding to (p,q) forms for q between 1 and n-2) \bar\partial_b cohomology groups of M with respect to L^2, Sobolev s, and smooth coefficients are finite and isomorphic to each other. The results are obtained by microlocalization using a new type of weight function called strongly CR plurisubharmonic.
math.CV↗
The \bar\partial_b Equation on Weakly Pseudoconvex CR Manifolds of Dimension 3
Let M be a smooth, compact, orientable, weakly pseudoconvex manifold of dimension 3, embedded in C^N (N greater than or equal to 2), of codimension one or more in C^N, and endowed with the induced CR structure. Assuming that the tangential Cauchy-Riemann operator \bar\partial_b has closed range in L^2 in order to rule out the Rossi example, we push regularity up to show \bar\partial_b has closed range in all Sobolev spaces s for s greater than zero. We then use the Szegö projection to show there is a smooth solution to the \bar\partial_b problem given smooth data. The results are obtained via microlocalization by piecing together estimates for functions and (0,1) forms that hold on different microlocal regions.
math.CV↗