Let K be a field of characteristic 0 and <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$k \ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> be an integer. We prove that every K-linear bijection <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$f : K[X] \rightarrow K[X]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <mi>K</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mi>K</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> strongly preserving the set of k-free polynomials (or the set of polynomials with a k-fold root in K) is a constant multiple of a K-algebra automorphism of K[X], i.e., there are elements <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$a, c \in K^{\times }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>,</mo> <mi>c</mi> <mo>∈</mo> <msup> <mi>K</mi> <mo>×</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$b \in K$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mo>∈</mo> <mi>K</mi> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$f(P)(X) = c P(a X + b)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>c</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mi>X</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. When K is a number field or <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$K=\mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>=</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>, we prove that similar statements hold when f preserves the set of polynomials with a root in K.

Symmetries of various sets of polynomials

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2025年7月12日

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Béranger Seguin

摘要

Let K be a field of characteristic 0 and $k \ge 2$ $k \geq 2$ be an integer. We prove that every K-linear bijection $f : K[X] \rightarrow K[X]$ $f : K [X] \to K [X]$ strongly preserving the set of k-free polynomials (or the set of polynomials with a k-fold root in K) is a constant multiple of a K-algebra automorphism of K[X], i.e., there are elements $a, c \in K^{\times }$ $a, c \in K^{\times}$ , $b \in K$ $b \in K$ such that $f(P)(X) = c P(a X + b)$ $f (P) (X) = c P (a X + b)$ . When K is a number field or $K=\mathbb {R}$ $K = R$ , we prove that similar statements hold when f preserves the set of polynomials with a root in K.