Suppose <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$C \subset {\mathbb {C}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mo>⊂</mo> <mi mathvariant="double-struck">C</mi> </mrow> </math></EquationSource> </InlineEquation> is compact. Let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$q_k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>q</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation> be a sequence of polynomials of degree <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n_k \rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>n</mi> <mi>k</mi> </msub> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, such that the locus of roots of all the polynomials is bounded, and the number of roots of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$q_k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>q</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation> in any closed set L disjoint from C is uniformly bounded. Supposing that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$(q_k)_k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>q</mi> <mi>k</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation> has an asymptotic root distribution <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> we provide conditions on C and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> assuring the sequence of mth derivatives <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$(q_k^{(m)})_k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>q</mi> <mi>k</mi> <mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation> has the same asymptotic root distribution <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> for any <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$m\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. This complements recent results of Totik, [<CitationRef CitationID="CR19">19</CitationRef>].

Zero Distributions of Derivatives of Polynomial Families Centering on a Set

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2026年3月7日

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Christian Henriksen,
Carsten Lunde Petersen,
Eva Uhre

摘要

Suppose $C \subset {\mathbb {C}}$ $C \subset C$ is compact. Let $q_k$ $q_{k}$ be a sequence of polynomials of degree $n_k \rightarrow \infty $ $n_{k} \to \infty$ , such that the locus of roots of all the polynomials is bounded, and the number of roots of $q_k$ $q_{k}$ in any closed set L disjoint from C is uniformly bounded. Supposing that $(q_k)_k$ ${(q_{k})}_{k}$ has an asymptotic root distribution $\mu $ $μ$ we provide conditions on C and $\mu $ $μ$ assuring the sequence of mth derivatives $(q_k^{(m)})_k$ ${(q_{k}^{(m)})}_{k}$ has the same asymptotic root distribution $\mu $ $μ$ for any $m\ge 1$ $m \geq 1$ . This complements recent results of Totik, [19].