TIFR CSE 2024 | Part B | Question: 11

Question

Let $\mathbb{C}$ denote the set of complex numbers and let $k$ be a positive integer. Given a non-zero univariate polynomial $f(x)$ with coefficients in $\mathbb{C}$ and an $a \in \mathbb{C}$, we say that $a$ is a zero of $f$ of multiplicity $k$ if $f(a)=0, \frac{d^{k} f}{d x^{k}}(a) \neq 0$, and for all $i \in\{1, \ldots, k-1\}$, $\frac{d^{i} f}{d x^{i}}(a)=0$.

Which of the following is true for every polynomial $f$ of degree $d$ and every positive integer $k$?

• A. The number of distinct zeroes in $\mathbb{C}$ of $f$ of multiplicity $k$ is at most $d / k$, and can be smaller than $d / k$ as well.
• B. The number of distinct zeroes in $\mathbb{C}$ of $f$ of multiplicity $k$ is at least $d / k$, and can be larger than $d / k$ as well.
• C. The number of distinct zeroes in $\mathbb{C}$ of $f$ of multiplicity $k$ is equal to $d / k$.
• D. The number of distinct zeroes in $\mathbb{C}$ of $f$ of multiplicity $k$ is at least $d$, and can be larger than $d$ as well.
• E. The number of distinct zeroes in $\mathbb{C}$ of $f$ of multiplicity $k$ is equal to $d$.