2017/09/27 by Andrew Chan, Andrew J. Chan, Diane Maclagan · 1 citation
Computer Science · Mathematics · #Polynomial and algebraic computation #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications
paper · pdf · doi:10.1090/mcom/3321
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a field with a valuation and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the polynomial ring <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S colon equals upper K left-bracket x 1 comma ellipsis comma x Subscript n Baseline right-bracket"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>:=</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">[</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">S:= K[x1, … , xn]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We discuss the extension of Gröbner theory to ideals in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , taking the valuations of coefficients into account, and describe the Buchberger algorithm in this context. In addition we discuss some implementation and complexity issues. The main motivation comes from tropical geometry, as tropical varieties can be defined using these Gröbner bases, but we also give examples showing that the resulting Gröbner bases can be substantially smaller than traditional Gröbner bases. In the case <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K equals double-struck upper Q"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">K =\mathbb Q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -adic valuation the algorithms have been implemented in a Macaulay 2 package.