Let A be a finite group (or a finite algebra), and <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation> be a word map (resp. polynomial map) on n many generators. We define the quantity <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$|\omega (A)|/|A|$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo stretchy="false">|</mo> <mo stretchy="false">/</mo> <mo stretchy="false">|</mo> <mi>A</mi> <mo stretchy="false">|</mo> </mrow> </math></EquationSource> </InlineEquation> as the image ratio of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation> on A and denote it by <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mu (\omega ,A)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>,</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. In this article, we investigate the set <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\textrm{R}(\omega )=\{\mu (\omega , A) : A {\text { is a finite group}}\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>R</mtext> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">{</mo> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>,</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>A</mi> <mrow> <mspace width="0.333333em" /> <mtext>is a finite group</mtext> </mrow> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>, and study the same for the case of rings. We demonstrate the existence of word maps whose set of image ratios is dense in [0, 1] for groups (and rings).

Image ratios of word maps and polynomial maps

期刊论文
发表日期：

2025年12月3日

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Saikat Panja

摘要

Let A be a finite group (or a finite algebra), and $\omega $ $ω$ be a word map (resp. polynomial map) on n many generators. We define the quantity $|\omega (A)|/|A|$ $| ω (A) | / | A |$ as the image ratio of $\omega $ $ω$ on A and denote it by $\mu (\omega ,A)$ $μ (ω, A)$ . In this article, we investigate the set $\textrm{R}(\omega )=\{\mu (\omega , A) : A {\text { is a finite group}}\}$ $R (ω) = {μ (ω, A) : A is a finite group}$ , and study the same for the case of rings. We demonstrate the existence of word maps whose set of image ratios is dense in [0, 1] for groups (and rings).