Questions tagged [zeroes]
The zeroes tag has no usage guidance.
48 questions
4
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Asymptotics of an entire function with real zeroes on the real line
Define $ F(x) := A x^m e^{B x} \prod_{k \geqslant 1} (1 - x/\alpha_k)e^{x/\alpha_k} $ where we suppose that $ \alpha_k \in \mathbb{R} $. This function is defined on the whole complex plane and is ...
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1
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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 ...
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1
vote
1
answer
106
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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 ...
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1
vote
1
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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
180
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Proof that the zeroes of certain polynomials are increasing with respect to degree
Choose $k+1$ positive integers $d_j\in\{0,1,2,3,\ldots\}$ and let $d=(d_1,\ldots,d_k)$.
Consider the following polynomial equation over the positive reals:
$$
\sum_{j=1}^{k}\frac1{x^{d_j}} = x^{d_{k+1}...
- 21
9
votes
1
answer
786
views
Sequence of real-rooted polynomials
I've been interested in proving a log-concavity result via proving that certain family of polynomials is real-rooted. By performing a sequence of transformations, I can reduce that problem to proving ...
- 1,889
2
votes
1
answer
206
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Bound for zero-crossings of heat equation
I am considering the following problem.
Let $\mathcal{P}$ the classical heat-diffusion problem:
$$\mathcal{P} : \left(\partial_t u (t,x)=\frac{1}{2}\partial_{xx}^2u(t,x)\text{ with }u(0,\cdot) = f(x)\...
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0
votes
1
answer
138
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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 ...
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2
votes
0
answers
420
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 ...
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3
votes
0
answers
152
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
219
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 ...
- 1,889
2
votes
0
answers
138
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 ...
- 233
7
votes
1
answer
193
views
Are entire functions uniformly bounded from below on a line through the origin?
Let $F : \mathbb C \to \mathbb C$ be an entire function of finite order. Since the zeros of $F$ are countable there exists a constant $c \in \mathbb R$ such that $F$ is zero-free on the line $e^{ic} \...
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7
votes
1
answer
410
views
Limit of zero sets of harmonic functions
Let $u_n : \mathbb{R}^n \to \mathbb{R}$ be a sequence of harmonic functions which converge uniformly on compact subsets. The limit function $u$ (which we assume to be not identically $0$) is clearly ...
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2
votes
1
answer
130
views
Zeroes of characters of general linear group induced from certain characters of parabolic subgroups
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Ind{Ind}$My question is about the types of conjugacy classes of $\GL(n,q)$, the general linear group over the finite field with $q$ elements, on which ...
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4
votes
0
answers
206
views
It is possible, without adding further hypotheses, to refine Rouche's theorem in order to obtain a finer localization of the zeros?
The title says it all: a now deleted question on the Mathematics Stackexchange asked more or less the same thing, and I answered by citing the work [1] of Wolfgang Tutschke, whose version of Rouche's ...
- 6,357
8
votes
0
answers
693
views
In need of help with parsing non-Archimedean function theory
My current work revolves around studying functions from the $p$-adic integers to the $q$-adic rationals, where $p$ and $q$ are distinct primes ("$(p,q)$-adic functions", as I call them). I'...
- 1,284
6
votes
2
answers
1k
views
Zero points of a smooth function on $\mathbb{R}$
Assume $f(x)$ is a smooth function on $\mathbb{R}$ and $f$ does not vanish on any interval. In other words, $f$ can have zero points but we cannot find any interval $(a, b)$ such that $f(x)=0$ for all ...
- 903
0
votes
1
answer
90
views
Change in the number of positive zeros of a continuous function
Let $f(x)$ be any continuous function, then is it true that $$Z^{+}\left(\alpha f(x)+(x+\beta)f'(x)\right)\leq Z^{+}\left(f'(x)\right)$$
where $\alpha>1$ is a real number and $\beta$ is any ...
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1
vote
0
answers
112
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A problem related to analytic function
Let, $z,w\in \mathbb{C}$. Let, $f(z)$ be an analytic function in $|z|<1$. Define, $f(z)= g(w)$ where $g(w)$ is analytic function in $\Re(w)>1/2$ and $w=\frac{1}{1+z^2}$ .
Question Prove that $$\...
user245973
3
votes
1
answer
757
views
Function whose sets of discontinuities and zeros are the rationals
Question: Is there a real valued function $f:\mathbb{R}\to\mathbb{R}$ such that its set of discontinuities is $\mathbb{Q}$ and its set of zeros $\{x\in \mathbb{R}\mid f(x)=0\}$ is also $\mathbb{Q}$?
...
- 2,160
14
votes
1
answer
448
views
Converse of the Lee-Yang circle theorem for polynomials with unitary roots
The Lee-Yang circle theorem states that if $\left( a_{ij} \right)$ is a Hermitian square $n \times n$ matrix whose entries are in the closed unit disc, then the polynomial $$ P\left(Z \right) = \sum_{...
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2
votes
1
answer
178
views
Unique continuation of the Hilbert transform
Let's consider the usual Hilbert transform $H$ defined as
$$Hf = P.V. (\frac{1}{x}*f).$$
A well-known unique continuation principle states that if $Hf = f =0$ on some interval $I$, then $f \equiv 0$. ...
- 903
3
votes
1
answer
398
views
Number of critical points of sum of two functions
I ran into the following "simple" question and I am wondering whether there are any references, which might help me. I am coming from statistics, so I am not so aware which branch of math ...
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4
votes
1
answer
101
views
Literature on the polynomials and equations, in structures with zero-divisors
I need literature about zeroes of polynomials and equation resolution in associative algebraic structures with zero-divisors, but I am having difficulties to find it.
For example, there is literature ...
2
votes
0
answers
233
views
Roots of determinant of matrix with polynomial entries — a generalization
For $1 \le i, j \le k$, consider $\rho_{ij}$ which are equal to either zero or one such that $\rho_{ii}=1$ and $\rho_{ij}=0$ if and only if $\rho_{ji}=0$. How to find the zeros of the determinant of ...
- 1,269
0
votes
0
answers
109
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The role of a combination of Eneström-Kakeya and Gauss-Lucas theorems: reference request or soft question, asking for this combination as tool
In past days I was trying to create problems or direct applications invoking Eneström—Kakeya and Gauss-Lucas theorems for certain arithmetic functions that I know from analytic number theory. These ...
1
vote
0
answers
182
views
What's the meaning of the nontrivial zeros of Selberg zeta function?
In the case of arithmetic variety over finite field, the zero points of the Hasse-Weil zeta function reflect the pure weights (i.e. dimension). On the other hand, in the case of the Selberg zeta ...
- 281
2
votes
1
answer
69
views
Dominant root of a family of odd degree polynomials
Let $q$ be an odd positive integer and denote by $f_q(x)=x^q-x^{q-2}-1$. This family of polynomials appears in some problems related to recurrence sequences.
In order to try to use some standard ...
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2
votes
1
answer
296
views
Homogeneous polynomial in 4 variable with non degenerate zero
I've got a very simple question about a homogenous polynomial, for which I cannot see neatly how to proceed (probably due to my limitations in algebraic geometry though). Any help would be greatly ...
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1
vote
0
answers
62
views
Example of an integer $n_0$ such that $1+\sum_{k=2}^{n_0} \zeta(k)^s=0$ has repeated roots
After I was studying the exercise Problem 4.20 from [1] I was inspired to ask about next problem, where $\zeta(k)$ denotes, for integers $k>1$, particular values of the Riemann zeta function. And $...
1
vote
0
answers
157
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
134
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)&...
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5
votes
1
answer
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Existence of Laurent series with zeroes at $𝑒^{2𝑛}$ ($𝑛∈ℕ_0$) and even faster coefficient decay
This is an extension of an earlier question of mine which corresponds to the case $A = 1$. Precisely, my question is as follows:
Given $A > 0$ fixed but arbitrary, is there a non-trivial sequence $...
- 734
9
votes
1
answer
412
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 ...
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38
votes
4
answers
4k
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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 ...
- 4,270
1
vote
1
answer
153
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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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1
vote
1
answer
194
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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 $\alpha \in [0,1]$...
- 1,719
0
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0
answers
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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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13
votes
2
answers
696
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 $C^\infty$-...
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-1
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1
answer
227
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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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7
votes
2
answers
661
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
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503
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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?
user111966
4
votes
2
answers
1k
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How to obtain an asymptotic formula for the zeros of the Airy function ($a_i$ for large $i$)?
Let $a_i$ be the zeros of the Airy function, which is the solution top the ODE $y''-xy=0$, such that Ai(a_i)=0. According to WolframMathWorld e.g., $a_{1..4}= -2.33811, -4.08795, -5.52056, -6....
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votes
1
answer
565
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Zeroes of a not quite holomorphic (but random if helpful) function
I’m interested in the zeroes of the complex function
$f(z,\bar{z}) = p(z) + \frac{1}{log(|z|)} q(z)$
where both $p$ and $q$ are polynomials of the complex variable $z$ (and are therefore holomorphic)...
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11
votes
3
answers
618
views
smooth functional to detect whether a function has a zero
Does there exist a function $F : C^\infty(\mathbb{R}, [0, \infty)) \to \mathbb{R}$ with the following properties:
$F(f) = 0$ if and only if there exists an $x \in [0,1]$ such that $f(x) = 0$.
$F$ is ...
- 1,092
2
votes
2
answers
206
views
Zeroes of trigonometric-like function
Consider a function $f(z)=\cos(z)\cosh(az)+\sin(z)\sinh(bz)$ for $z\in \mathbb{C}, a,b \in \mathbb{R}$. Denote $D\subseteq \mathbb{R}^2$ being the set of such pairs $(a,b)$ of parameters so that NOT ...
- 1,550
1
vote
0
answers
532
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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.
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