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• W3214070287 abstract "Algorithms working with linear algebraic groups often represent them via defining polynomial equations. One can always choose defining equations for an algebraic group to be of degree at most the degree of the group as an algebraic variety. However, the degree of a linear algebraic group <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G subset-of upper G upper L Subscript n Baseline left-parenthesis upper C right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>⊂<!-- ⊂ --></mml:mo> <mml:msub> <mml:mi>GL</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>C</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>G subset operatorname {GL}_n(C)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be arbitrarily large even for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n equals 1> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>n = 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. One of the key ingredients of Hrushovski’s algorithm for computing the Galois group of a linear differential equation was an idea to “approximate” every algebraic subgroup of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G upper L Subscript n Baseline left-parenthesis upper C right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>GL</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>C</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>operatorname {GL}_n(C)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by a “similar” group so that the degree of the latter is bounded uniformly in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=application/x-tex>n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Making this uniform bound computationally feasible is crucial for making the algorithm practical. In this paper, we derive a single-exponential degree bound for such an approximation (we call it a <italic>toric envelope</italic>), which is qualitatively optimal. As an application, we improve the quintuply exponential bound due to Feng for the first step of Hrushovski’s algorithm to a single-exponential bound. For the cases <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n equals 2 comma 3> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>n = 2, 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> often arising in practice, we further refine our general bound." @default.
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• W3214070287 date "2021-10-28" @default.
• W3214070287 modified "2023-10-15" @default.
• W3214070287 title "Degree bound for toric envelope of a linear algebraic group" @default.
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• W3214070287 doi "https://doi.org/10.1090/mcom/3695" @default.
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