# Questions tagged [zeroes]

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### Function whose sets of discontinuities and zeros are the rationals

Question: Is there a real valued function $f:\mathbb{R}\to\mathbb{R}$ such that its set of discontinuities is $\mathbb{Q}$ and its set of zeros $\{x\in \mathbb{R}\mid f(x)=0\}$ is also $\mathbb{Q}$? ...
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### Existence of Laurent series with zeroes at $e^{2n}$ ($n \in \Bbb{N}_0$) and extremely fast coefficient decay

I am working on a problem in harmonic analysis, which I converted into the following existence problem concerning Laurent series. I am a bit at a loss concerning this problem, since my knowledge of ...
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### A family of polynomials whose zeros all lie on the unit circle

I had posted the following problem on stack exchange before. Suppose $\lambda$ is a real number in $\left( 0,1\right)$, and let $n$ be a positive integer. Prove that all the roots of the ...
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### Is there always a polynomial with real zeroes between two polynomials with real zeroes?

Suppose that we have two complex polynomials $p(z)=\sum_{k=0}^n p_kz^k$ and $q(z)=\sum_{k=0}^n q_kz^k$ and also that we have $|p_k|<|q_k|$ for $k=0,1,...,n$. We say that a polynomial $r$ is ...
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### Locus of roots of all convex combinations of two monic polynomials, II

This post contains a revised conjecture to a conjecture I posed previously which was shown to be false. Let $p, q \in \mathbb{C}[t]$ be two monic polynomials of degree $n \ge 1$. For $\alpha \in [0,1]$...
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### Deriving “quasi-theta” functions from theta functions' zeros

I've been trying to sift the zeros of \begin{align}\vartheta(z)&=e^{-\pi z^2}\prod_{k\ge1}(1+e^{-(2k-1)\pi+2\pi z})(1+e^{-(2k-1)\pi-2\pi z})\\ &=\prod_{k\ge1}(1+e^{-(2k-1)\pi+2i\pi z})(1+e^{-(...
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### The $n$-th derivative has $n$ zeros. Can such a function be unbounded?

I asked this on Math.SE some days ago, but without any success. For some application I need a formal definition of bell-shaped function. So I had the following idea: Definition. A $C^\infty$-...
161 views

### A simple question about the zeros of an Entire Function in LP-class

We know that functions in $\mathcal{LP}$-class, and only these, are uniform limits, on compact subsets of $\mathbb{C}$, of polynomials with only real zeros. Question: Does it means that these ...
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### Locus of roots of all convex combinations of two monic polynomials

Let $p,q$ be monic polynomials in $\mathbb{C}[t]$ and for $\alpha \in [0,1]$, let $c_\alpha := \alpha p + (1-\alpha)q \in \mathbb{C}[t]$. Since the roots of a polynomial vary continuously with respect ...
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### Are trivial zeros of the zeta function important?

Non-trivial zeros play an important (main) role in the distribution of prime numbers. Are there theorems in which trivial zeros play an important (main) role?