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• W3105265498 abstract "An old conjecture of Erdős and Rényi, proved by Schinzel, predicted a bound for the number of terms of a polynomial <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=g left-parenthesis x right-parenthesis element-of double-struck upper C left-bracket x right-bracket> <mml:semantics> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>C</mml:mi> </mml:mrow> <mml:mo stretchy=false>[</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>]</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>g(x)in mathbb {C}[x]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> when its square <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=g left-parenthesis x right-parenthesis squared> <mml:semantics> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>g(x)^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has a given number of terms. Further conjectures and results arose, but some fundamental questions remained open. In this paper, with methods which appear to be new, we achieve a final result in this direction for completely general algebraic equations <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f left-parenthesis x comma g left-parenthesis x right-parenthesis right-parenthesis equals 0> <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>g</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>f(x,g(x))=0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <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> <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:mrow> <mml:annotation encoding=application/x-tex>f(x,y)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is monic of arbitrary degree in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=y> <mml:semantics> <mml:mi>y</mml:mi> <mml:annotation encoding=application/x-tex>y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and has boundedly many terms in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=x> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding=application/x-tex>x</mml:annotation> </mml:semantics> </mml:math> </inline-formula>: we prove that the number of terms of such a <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=g left-parenthesis x right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>g(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is necessarily bounded. This includes the previous results as extremely special cases. We shall interpret polynomials with boundedly many terms as the restrictions to 1-parameter subgroups or cosets of regular functions of bounded degree on a given torus <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper G Subscript m Superscript l> <mml:semantics> <mml:msubsup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>G</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mtext>m</mml:mtext> </mml:mrow> <mml:mi>l</mml:mi> </mml:msubsup> <mml:annotation encoding=application/x-tex>mathbb {G}_textrm {m}^l</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Such a viewpoint shall lead to some best-possible corollaries in the context of finite covers of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper G Subscript m Superscript l> <mml:semantics> <mml:msubsup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>G</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mtext>m</mml:mtext> </mml:mrow> <mml:mi>l</mml:mi> </mml:msubsup> <mml:annotation encoding=application/x-tex>mathbb {G}_textrm {m}^l</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, concerning the structure of their integral points over function fields (in the spirit of conjectures of Vojta) and a Bertini-type irreducibility theorem above algebraic multiplicative cosets. A further natural reading occurs in non-standard arithmetic, where our result translates into an algebraic and integral-closedness statement inside the ring of non-standard polynomials." @default.
• W3105265498 created "2020-11-23" @default.
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• W3105265498 date "2017-03-01" @default.
• W3105265498 modified "2023-10-08" @default.
• W3105265498 title "On fewnomials, integral points, and a toric version of Bertini’s theorem" @default.
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• W3105265498 doi "https://doi.org/10.1090/jams/878" @default.
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