Fewnomial systems with many roots, and an Adelic Tau Conjecture Conference Paper

abstract

• Consider a system F of n polynomials in n variables, with a total of n+k distinct exponent vectors, over any local field L. We discuss conjecturally tight bounds on the maximal number of non-degenerate roots F can have over L, with all coordinates having fixed phase, as a function of n, k, and L only. In particular, we give new explicit systems with number of roots approaching the best known upper bounds. We also briefly review the background behind such bounds, and their application, including connections to computational number theory and variants of the Shub-Smale tau-Conjecture and the P vs. NP Problem. One of our key tools is the construction of combinatorially constrained tropical varieties with maximally many intersections.

authors

• Rojas, Joseph

published proceedings

• TROPICAL AND NON-ARCHIMEDEAN GEOMETRY

author list (cited authors)

• Phillipson, K., & Rojas, J. M.

citation count

• 8

complete list of authors

• Phillipson, Kaitlyn||Rojas, J Maurice

publication date

• November 2013

publisher

• American Mathematical Society (AMS)  Publisher

published in

• Contemporary Mathematics  Journal

keywords

• Complexity
• Local Field
• Lower Bounds
• Mixed Cell
• Positive Characteristic
• Sparse Polynomial
• Straight-line Program
• Tau Conjecture

Digital Object Identifier (DOI)

• 10.1090/conm/605/12111

International Standard Book Number (ISBN) 13

• 978-1-4704-1021-6

start page

• 45

end page

• 71

volume

• 605

URL

• http%3A%2F%2Fdx.doi.org%2F10.1090%2Fconm%2F605%2F12111