Multiplicative series, modular forms, and Mandelbrot polynomials

Abstract

We say a power series <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript n equals 0 Overscript normal infinity Endscripts a Subscript n Baseline q Superscript n"> <mml:semantics> <mml:mrow> <mml:munderover> <mml:mo>∑<!-- ∑ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:munderover> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:msup> <mml:mi>q</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\sum _{n=0}^\infty a_n q^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <italic>multiplicative</italic> if the sequence <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 comma a 2 slash a 1 comma ellipsis comma a Subscript n Baseline slash a 1 comma ellipsis"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mi>a</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>a</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…<!-- … --></mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">1,a_2/a_1,\ldots ,a_n/a_1,\ldots</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is so. In this paper, we consider multiplicative power series <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f squared"> <mml:semantics> <mml:msup> <mml:mi>f</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">f^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is also multiplicative. We find a number of examples for which <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a rational function or a theta series and prove that the complete set of solutions is the locus of a (probably reducible) affine variety over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The precise determination of this variety turns out to be a finite computational problem, but it seems to be beyond the reach of current computer algebra systems. The proof of the theorem depends on a bound on the logarithmic capacity of the Mandelbrot set.

Locations

• Mathematics of Computation - View
• arXiv (Cornell University) - View - PDF