grandes-ecoles 2025 Q4

grandes-ecoles · France · mines-ponts-maths1__psi Roots of polynomials Reciprocal and antireciprocal polynomial properties

Let $R$ be a non-constant polynomial of $\mathbf{C}[X]$ having the following property: Every root $a$ of $R$ is nonzero and $\frac{1}{a}$ is a root of $R$ with the same multiplicity as $a$. Prove that the product of the roots of $R$, counted with multiplicities, can only take the values 1 or $-1$. One may note that the equality $a = \frac{1}{a}$ holds only for $a = 1$ or $-1$.

Let $R$ be a non-constant polynomial of $\mathbf{C}[X]$ having the following property: Every root $a$ of $R$ is nonzero and $\frac{1}{a}$ is a root of $R$ with the same multiplicity as $a$.\\
Prove that the product of the roots of $R$, counted with multiplicities, can only take the values 1 or $-1$. One may note that the equality $a = \frac{1}{a}$ holds only for $a = 1$ or $-1$.

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