Tropical Effective Primary and Dual Nullstellens\"atze

Tropical algebra is an emerging field with a number of applications in various areas of mathematics. In many of these applications appeal to tropical polynomials allows to study properties of mathematical objects such as algebraic varieties and algebraic curves from the computational point of view. This makes it important to study both mathematical and computational aspects of tropical polynomials. In this paper we prove a tropical Nullstellensatz and moreover we show an effective formulation of this theorem. Nullstellensatz is a natural step in building algebraic theory of tropical polynomials and its effective version is relevant for computational aspects of this field. On our way we establish a simple formulation of min-plus and tropical linear dualities. We also observe a close connection between tropical and min-plus polynomial systems

Yao’s millionaires’ problem and public-key encryption without computational assumptions

We offer efficient and practical solutions of Yao’s millionaires’ problem without using any one-way functions. Some of the solutions involve physical principles, while others are purely mathematical. One of our solutions (based on physical principles) yields a public-key encryption protocol secure against (passive) computationally unbounded adversary. In that protocol, the legitimate parties are not assumed to be computationally unbounded. </jats:p

A Monte Carlo Test of the Optimal Jet Definition

We summarize the Optimal Jet Definition and present the result of a benchmark Monte Carlo test based on the W-boson mass extraction from fully hadronic decays of pairs of W's.Comment: 7 pages, talk given at Lake Louise Winter Institute: "Particles and the Universe", Lake Louise, Canada, February 16-22, 2003, to be published in the proceeding