Skip to main content

Questions tagged [zeroes]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
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 ...
user528052's user avatar
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 ...
Math101's user avatar
  • 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 ...
chrisv's user avatar
  • 21
0 votes
0 answers
178 views

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}...
chrisv's user avatar
  • 21
9 votes
1 answer
696 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 ...
Luis Ferroni's user avatar
  • 1,949
2 votes
1 answer
201 views

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)\...
NancyBoy's user avatar
  • 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 ...
Bobby Ocean's user avatar
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 ...
VSP's user avatar
  • 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 $\sigma>0$ ...
NancyBoy's user avatar
  • 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 ...
Luis Ferroni's user avatar
  • 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 ...
Desura's user avatar
  • 171
7 votes
1 answer
182 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} \...
J. Swail's user avatar
  • 437
7 votes
1 answer
389 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 ...
user492517's user avatar
2 votes
1 answer
119 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 ...
mathseeker's user avatar
4 votes
0 answers
178 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 ...
Daniele Tampieri's user avatar
8 votes
0 answers
671 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'...
MCS's user avatar
  • 1,274
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 ...
Jacob Lu's user avatar
  • 903
0 votes
1 answer
89 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 ...
VSP's user avatar
  • 233
1 vote
0 answers
110 views

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 $$\...
user avatar
3 votes
1 answer
749 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 $\mathbb{Q}$? ...
tj_'s user avatar
  • 2,170
14 votes
1 answer
435 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_{...
darkl's user avatar
  • 680
2 votes
1 answer
162 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 $f \equiv 0$. ...
Jacob Lu's user avatar
  • 903
3 votes
1 answer
357 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 ...
BrainSlap's user avatar
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 ...
Dragon lala lalo's user avatar
2 votes
0 answers
232 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 ...
GA316's user avatar
  • 1,229
0 votes
0 answers
106 views

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 ...
user142929's user avatar
1 vote
0 answers
180 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 ...
elliptic curve's user avatar
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 ...
Lennon's user avatar
  • 21
2 votes
1 answer
267 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 ...
Gil Sanders's user avatar
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 $...
user142929's user avatar
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 ...
user142929's user avatar
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)&...
Antonio DJC's user avatar
0 votes
0 answers
110 views

Root of the expectation of a random rational function

I am trying to figure out a formula for the unique $\lambda>1$ such that $$ \mathbb{E}\bigg[\frac{X}{\lambda -X}\bigg]=1 $$ where $X$ is a discrete random variable taking values in $\{\frac{1}{n},....
Gabriel's user avatar
  • 29
5 votes
1 answer
168 views

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 $...
PhoemueX's user avatar
  • 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 ...
PhoemueX's user avatar
  • 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 ...
math110's user avatar
  • 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 ...
Shalom's user avatar
  • 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 $\alpha \in [0,1]$...
Pietro Paparella's user avatar
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^{-(...
Antonio DJC's user avatar
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 $C^\infty$-...
M. Winter's user avatar
  • 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 ...
Dennis Jia's user avatar
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 ...
Pietro Paparella's user avatar
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?
user avatar
4 votes
2 answers
997 views

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....
user1611107's user avatar
5 votes
1 answer
561 views

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)...
user41147's user avatar
  • 263
11 votes
3 answers
614 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 ...
Dan Christensen's user avatar
2 votes
2 answers
201 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 ...
Sergei's user avatar
  • 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.
user21239's user avatar