Minimal Castelnuovo-Mumford regularity for a given Hilbert polynomial

Let $K$ be an algebraically closed field of null characteristic and $p(z)$ a Hilbert polynomial. We look for the minimal Castelnuovo-Mumford regularity $m_{p(z)}$ of closed subschemes of projective spaces over $K$ with Hilbert polynomial $p(z)$. Experimental evidences led us to consider the idea that $m_{p(z)}$ could be achieved by schemes having a suitable minimal Hilbert function. We give a constructive proof of this fact. Moreover, we are able to compute the minimal Castelnuovo-Mumford regularity $m_p(z)^{\varrho}$ of schemes with Hilbert polynomial $p(z)$ and given regularity $\varrho$ of the Hilbert function, and also the minimal Castelnuovo-Mumford regularity $m_u$ of schemes with Hilbert function $u$. These results find applications in the study of Hilbert schemes. They are obtained by means of minimal Hilbert functions and of two new constructive methods which are based on the notion of growth-height-lexicographic Borel set and called ideal graft and extended lifting.Comment: 21 pages. Comments are welcome. More concise version with a slight change in the title. A further revised version has been accepted for publication in Experimental Mathematic

Proof of the Dubrovin conjecture and analysis of the tritronqu\'ee solutions of PIP_I

We show that the tritronqu\'ee solution of the Painlev\'e equation $\P1$, $y"=6y^2+z$ which is analytic for large $z$ with $\arg z \in (-\frac{3\pi}{5}, \pi)$ is pole-free in a region containing the full sector ${z \ne 0, \arg z \in [-\frac{3\pi}{5}, \pi]}$ and the disk ${z: |z| < 37/20}$. This proves in particular the Dubrovin conjecture, an open problem in the theory of Painlev\'e transcendents. The method, building on a technique developed in Costin, Huang, Schlag (2012), is general and constructive. As a byproduct, we obtain the value of the tritronqu\'ee and its derivative at zero within less than 1/100 rigorous error bounds

The Abhyankar-Jung Theorem

We show that every quasi-ordinary Weierstrass polynomial P(Z) = Z^d+a_1 (X) Z^{d-1}+...+a_d(X) \in \K[[X]][Z] , $X=(X_1,..., X_n)$, over an algebraically closed field of characterisic zero \K, and satisfying $a_1=0$, is $\nu$-quasi-ordinary. That means that if the discriminant \Delta_P \in \K[[X]] is equal to a monomial times a unit then the ideal $(a_i^{d!/i}(X))_{i=2,...,d}$ is principal and generated by a monomial. We use this result to give a constructive proof of the Abhyankar-Jung Theorem that works for any Henselian local subring of \K[[X]] and the function germs of quasi-analytic families.Comment: 14 pages. The toric case has been added. To be published in Journal of Algebr