scholarly journals Detecting tropical defects of polynomial equations

2019 ◽  
Author(s):  
Paul Görlach ◽  
Yue Ren ◽  
Jeff Sommars
Keyword(s):  
Del Pezzo Surfaces ◽  
Polynomial Equations ◽  
Zero Set ◽  
Open Problems ◽  
Affine Spaces ◽  
Del Pezzo ◽  
System Of Polynomial Equations ◽  
Tropical Basis

Abstract We introduce the notion of tropical defects, certificates that a system of polynomial equations is not a tropical basis, and provide two algorithms for finding them in affine spaces of complementary dimension to the zero set. We use these techniques to solve open problems regarding del Pezzo surfaces of degree 3 and realizability of valuated gaussoids on 4 elements.

scholarly journals On Reducing a System of Equations to a Single Equation

2004 ◽  
Vol 11 (6) ◽  
Author(s):  
Gudmund Skovbjerg Frandsen ◽  
Igor E. Shparlinski
Keyword(s):  
Original System ◽  
Single Equation ◽  
Small Degree ◽  
System Of Equations ◽  
Polynomial Equations ◽  
Zero Set ◽  
Theoretic Result ◽  
Straight Line ◽  
Efficient Construction ◽  
System Of Polynomial Equations

For a system of polynomial equations over Q_p we present an efficient construction of a single polynomial of quite small degree whose zero set over Q_p coincides with the zero set over Q_p of the original system. We also show that the polynomial has some other attractive features such as low additive and straight-line complexity.<br /> <br /> The proof is based on a link established here between the above problem and some recent number theoretic result about zeros of p-adic forms.

scholarly journals Cox rings of degree one del Pezzo surfaces

2009 ◽  
Vol 3 (7) ◽  
pp. 729-761 ◽  
Author(s):  
Damiano Testa ◽  
Anthony Várilly-Alvarado ◽  
Mauricio Velasco
Keyword(s):  
Del Pezzo Surfaces ◽  
Cox Rings ◽  
Del Pezzo

scholarly journals Tropicalization of del Pezzo surfaces

2016 ◽  
Vol 300 ◽  
pp. 156-189 ◽  
Author(s):  
Qingchun Ren ◽  
Kristin Shaw ◽  
Bernd Sturmfels
Keyword(s):  
Del Pezzo Surfaces ◽  
Del Pezzo

scholarly journals Manin’s conjecture for quartic Del Pezzo surfaces with a conic fibration

2011 ◽  
Vol 160 (1) ◽  
pp. 1-69 ◽  
Author(s):  
R. De la Bretèche ◽  
T. D. Browning
Keyword(s):  
Del Pezzo Surfaces ◽  
Del Pezzo ◽  
Manin’S Conjecture

scholarly journals Weak del Pezzo surfaces with irregularity

2007 ◽  
Vol 59 (2) ◽  
pp. 293-322 ◽  
Author(s):  
Stefan SchrÖer
Keyword(s):  
Del Pezzo Surfaces ◽  
Del Pezzo
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