인공고관절의 세라믹 볼 헤드의 기계적 안정성 평가를 위한 3 차원 유한요소 해석

Convex bodies: the brunn–minkowski theory

Smale’s α-theory uses estimates related to the convergence of Newton’s method to certify that Newton iterations will converge quadratically to solutions to a square polynomial system. The program alphaCertified implements algorithms based on α-theory to certify solutions of polynomial systems using both exact rational arithmetic and arbitrary precision floating point arithmetic. It also implements algorithms that certify whether a given point corresponds to a real solution, and algorithms to heuristically validate solutions to overdetermined systems. Examples are presented to demonstrate the algorithms.

/pdf/algorithm-921-alphacertified-certifying-solutions-to-4yb5vrz9u7.pdf

Algorithm 921: alphaCertified: Certifying Solutions to Polynomial Systems

Searching for structural reasons behind old results and conjectures of Chudnovksy regarding the least degree of a nonzero form in an ideal of fat points in projective N-space, we make conjectures which explain them, and we prove the conjectures in certain cases, including the case of general points in the projective plane. Our conjectures were also partly motivated by the Eisenbud-Mazur Conjecture on evolutions, which concerns symbolic squares of prime ideals in local rings, but in contrast we consider higher symbolic powers of homogeneous ideals in polynomial rings.

Are symbolic powers highly evolved

We develop tools to study the problem of containment of symbolic powers $I^{(m)}$ in powers $I^r$ for a homogeneous ideal $I$ in a polynomial ring $k[{\bf P}^N]$ in $N+1$ variables over an algebraically closed field $k$. We obtain results on the structure of the set of pairs $(r,m)$ such that $I^{(m)}\subseteq I^r$. As corollaries, we show that $I^2$ contains $I^{(3)}$ whenever $S$ is a finite generic set of points in ${\bf P}^2$ (thereby giving a partial answer to a question of Huneke), and we show that the containment theorems of Ein-Lazarsfeld-Smith and Hochster-Huneke are optimal for every fixed dimension and codimension.

Comparing powers and symbolic powers of ideals

Seshadri constants express the so called local positivity of a line bundle on a projective variety. They were introduced by Demailly. The original idea of using them towards a proof of the Fujita conjecture failed but they quickly became a subject of intensive study quite in their own right. Lazarsfeld's book "Positivity in Algebraic Geometry" contains a whole chapter devoted to local positivity and serves as a very enjoyable introduction to Seshadri constants. Since this book has appeared, the subject witnessed quite a bit of development. It is the aim of these notes to give an account of recent progress as well as to discuss many open questions and provide some examples.

A primer on Seshadri constants

Seshadri constants express the so called local positivity of a line bundle on a projective variety. They were introduced in [Dem92] by Demailly. The original hope of using them towards a proof of t ...

/pdf/a-primer-on-seshadri-constants-16ibtlgnof.pdf

While classical algebraic geometry has been studied for hundreds of years, numerical algebraic geometry has only recently been developed. Due in large part to the work of Andrew Sommese and his collaborators, the intersection of these two fields is now ripe for rapid advancement. The primary goal of both the conference and this volume is to foster the interaction between researchers interested in classical algebraic geometry and those interested in numerical methods.

Interactions of Classical and Numerical Algebraic Geometry

A k-very ample line bundle L on a Del Pezzo Surface is numerically characterized, improving the results of Biancofiore— Ceresa in [7].

k—Very Ample Line Bundles on Del Pezzo Surfaces

Non-singular toric embeddings with dual defect are classified. The associated polytopes, called defect polytopes, are proven to be the class of Delzant integral polytopes for which a combinatorial invariant vanishes. The structure of a defect polytope is described.

Sandra Di Rocco

Papers

A primer on Seshadri constants

A primer on Seshadri constants

Interactions of Classical and Numerical Algebraic Geometry

k—Very Ample Line Bundles on Del Pezzo Surfaces

Projective Duality of Toric Manifolds and Defect Polytopes