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- W2964319398 abstract "A pair of symmetric bilinear forms <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=application/x-tex>A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper B> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding=application/x-tex>B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> determine a binary form <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f left-parenthesis x comma y right-parenthesis colon equals d i s c left-parenthesis upper A x minus upper B y right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>:=</mml:mo> <mml:mi>disc</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>A</mml:mi> <mml:mi>x</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mi>B</mml:mi> <mml:mi>y</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>f(x,y) := operatorname {disc}(Ax-By)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We prove that the question of whether a given binary form can be written in this way as a discriminant form generically satisfies a local-global principle and deduce from this that most binary forms over <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper Q> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>Q</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are discriminant forms. This is related to the arithmetic of the hyperelliptic curve <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=z squared equals f left-parenthesis x comma y right-parenthesis> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>z</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>z^2 = f(x,y)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Analogous results for nonhyperelliptic curves are also given." @default.
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- W2964319398 date "2016-12-06" @default.
- W2964319398 modified "2023-09-26" @default.
- W2964319398 title "Most binary forms come from a pencil of quadrics" @default.
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- W2964319398 doi "https://doi.org/10.1090/bproc/24" @default.
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