Bounding the radii of balls meeting every connected component of semi-algebraic sets

We prove explicit bounds on the radius of a ball centered at the origin which is guaranteed to contain all bounded connected components of a semi-algebraic set S \subset \mathbbm{R}^k defined by a quantifier-free formula involving $s$ polynomials in \mathbbm{Z}[X_1, ..., X_k] having degrees at most $d$, and whose coefficients have bitsizes at most $\tau$. Our bound is an explicit function of $s, d, k$ and $\tau$, and does not contain any undetermined constants. We also prove a similar bound on the radius of a ball guaranteed to intersect every connected component of $S$ (including the unbounded components). While asymptotic bounds of the form $2^{\tau d^{O (k)}}$ on these quantities were known before, some applications require bounds which are explicit and which hold for all values of $s, d, k$ and $\tau$. The bounds proved in this paper are of this nature.Comment: 11 page

Geometric invariant theory approach to the determination of ground states of D-wave condensates in isotropic space

A complete and rigorous determination of the possible ground states for D-wave pairing Bose condensates is presented, using a geometrical invariant theory approach to the problem. The order parameter is argued to be a vector, transforming according to a ten dimensional real representation of the group $G=${\bf O}$_3\otimes${\bf U}$_1\times$. We determine the equalities and inequalities defining the orbit space of this linear group and its symmetry strata, which are in a one-to-one correspondence with the possible distinct phases of the system. We find 15 allowed phases (besides the unbroken one), with different symmetries, that we thoroughly determine. The group-subgroup relations between bordering phases are pointed out. The perturbative sixth degree corrections to the minimum of a fourth degree polynomial $G$-invariant free energy, calculated by Mermin, are also determined.Comment: 27 revtex pages, 2 figures, use of texdraw; minor changes in the bibliography and in Table II

Moduli Spaces of Semistable Sheaves on Singular Genus One Curves

We find some equivalences of the derived category of coherent sheaves on a Gorenstein genus one curve that preserve the (semi)-stability of pure dimensional sheaves. Using them we establish new identifications between certain Simpson moduli spaces of semistable sheaves on the curve. For rank zero, the moduli spaces are symmetric powers of the curve whilst for a fixed positive rank there are only a finite number of non-isomorphic spaces. We prove similar results for the relative semistable moduli spaces on an arbitrary genus one fibration with no conditions either on the base or on the total space. For a cycle $E_N$ of projective lines, we show that the unique degree 0 stable sheaves are the line bundles having degree 0 on every irreducible component and the sheaves $\mathcal{O}(-1)$ supported on one irreducible component. We also prove that the connected component of the moduli space that contains vector bundles of rank $r$ is isomorphic to the $r$-th symmetric product of the rational curve with one node.Comment: 26 pages, 4 figures. Added the structure of the biggest component of the moduli space of sheaves of degree 0 on a cycle of projective lines. Final version; to appear en IMRS (International Mathematics Research Notices 2009