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# Questions tagged [zeroes]

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5 votes
1 answer
353 views

### What is the length of an algebraic curve?

The following question seems to be somewhat standard, but I was unable to find any reference. I would be grateful for any pointers to relevant literature. We consider a real polynomial $p(x,y)$ of ...
1 vote
1 answer
103 views

### Characterizing the unimodular functions from the closed disk $\overline{\mathbb{D}}$ to $\mathbb{C}$ with constraints

Let $\mathbb{D}$ be the open disc. It is well known that if $f:\mathbb{D}\to\mathbb{C}$ is analytic, continuous on the boundary, and is unimodular (say with a finite number of zeros) then $f$ is a ...
• 167
1 vote
1 answer
233 views

### Zeroes of elementary polynomials without involving closed-form solutions

Consider the following two polynomials, where $n$ is an integer: $$p_n(x) = x^3-\frac1nx-\frac2n, \\ q_n(x) = x^2-\frac2n.$$ For any $n$, let $x_p=x_p(n)$ and $x_q=x_q(n)$ be the unique positive ...
• 21
0 votes
0 answers
178 views

• 393
0 votes
1 answer
131 views

### Is $\Gamma(z,1)\not=0$ for all $z$ with $\Re(z)<0$?

I found this paper online which appears to present zeros of the incomplete gamma function within the right half plane. It makes me think that there are no zeros in the left half plane. Not sure how to ...
• 410
2 votes
0 answers
313 views

### Functional continuity of eigenvalues?

We have the following theorems! Corollary VI.1.6 [Bhatia: Matrix Analysis]: Let $a_j(t)$, where $1\leq j \leq n$ be continuous complex valued functions defined on an interval $I$. Then there exists ...
• 233
3 votes
0 answers
146 views

### Inequality involving convolution roots

I am struggling with the following problem. Let $f$ be a real smooth function. Let assume that $f$ is: increasing strictly convex on $(-\infty,0)$ strictly concave on $(0,+\infty)$ Let $\sigma>0$ ...
• 393
7 votes
0 answers
212 views

### Partitions, weights and polynomials with roots on the unit circle

Let us consider the set $[n]=\{1,\ldots,n\}$ and all of its partitions into exactly $m$ blocks, but let us allow each block to be internally ordered. For example, taking $n=6$ and $m=2$, we will ...
• 1,949
2 votes
0 answers
130 views

### On the solutions of $_2F_1(\alpha, \beta; \gamma, z) = \Lambda$

More General Question Let $$F(\alpha,\beta;\gamma;z) = \sum_{n=0}^{+\infty} \frac{(\alpha)_n(\beta)_n}{(\gamma)_nn!}z^n, \quad |z| < 1, \quad (x)_n = x(x+1)...(x+n-1), \quad (x)_0 = 1$$ be the ...
• 171
7 votes
1 answer
182 views

1 vote
0 answers
156 views

### Solution of $s\zeta'(s)-\zeta(s)=0$ for some infinite strip $\subset\{0<\Re s<1\}$, with $\Im s>0$

I did the following calculation, first I take the $k$-th derivative with respect $s$ of the Mellin transform of the fractional part $\{\frac{1}{t}\}$ defined for $0<\Re s<1$ as it is showed in ...
3 votes
0 answers
127 views

### An elliptic function built from a log-theta-function integral?

I'm studying the apparent ellipticity of $$\Theta(z,a):=\frac{\exp(\tfrac2a\int_0^a\ln\vartheta(x+z)\,dx)}{\vartheta(z)\vartheta(z+a)},\tag1$$with $a$ is a free parameter and \begin{align}\vartheta(z)&...
• 149
0 votes
0 answers
110 views

• 754
9 votes
1 answer
399 views

### 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 ...
• 754
38 votes
4 answers
4k views

### 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 ...
• 4,270
1 vote
1 answer
151 views

### 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 ...
• 513
1 vote
1 answer
194 views

### 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 $\alpha \in [0,1]$...
• 1,719
0 votes
0 answers
83 views

### 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^{-(...
• 149
13 votes
2 answers
683 views

### 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 $C^\infty$-...
• 12.8k
-1 votes
1 answer
221 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 ...
7 votes
2 answers
657 views

### 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 ...
• 1,719
8 votes
2 answers
500 views

### 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?
4 votes
2 answers
997 views

• 1,550
1 vote
0 answers
532 views

### How to find all the zeros of a cubic spline?

Let's say I have a cubic spline represented piecewise by cubic polynomials. Do you know an efficient algorithm for computing all its zeros? Thank you.