2013/01/31 by Martin Bays, Philipp Habegger · 4 citations
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Algebraic curve #Algebraic number #Commutative Algebra and Its Applications #Conjecture #Dimension (graph theory) #Field (mathematics) #Geometry #Mathematical analysis #Mathematics #Polynomial and algebraic computation #Pure mathematics #Torus #Transcendence (philosophy) #math.LO #math.NT #msc:03C64 #msc:11G50 #msc:11J86 #msc:11U09 #msc:14H25
paper · pdf · doi:10.1090/s0002-9947-2014-06494-5
published in Transactions of the American Mathematical Society 367(2), 1313-1328 (American Mathematical Society) · Published version, but with an error fixed in the formula for the function on page 2
openalex publication_date 2014/09/04 · arxiv created 2015/04/21 · arxiv updated 2015/04/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C"> <mml:semantics> <mml:mi>C</mml:mi> <mml:annotation encoding="application/x-tex">C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an irreducible algebraic curve defined over a number field and inside an algebraic torus of dimension at least <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We partially answer a question posed by Levin on points on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C"> <mml:semantics> <mml:mi>C</mml:mi> <mml:annotation encoding="application/x-tex">C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which a non-trivial power lies again on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C"> <mml:semantics> <mml:mi>C</mml:mi> <mml:annotation encoding="application/x-tex">C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Our results have connections to Zilber’s Conjecture on Intersections with Tori and yield to methods arising in transcendence theory and the theory of o-minimal structures.