# Questions tagged [exponential-polynomials]

The exponential-polynomials tag has no usage guidance.

38
questions

**-1**

votes

**0**answers

43 views

### When is $f_r(x)$ is identically zero?

For $r \in \mathbf{R},$ let $$f_r(x) = \sum_{j=1}^{n} c_j (x+p_j)^r$$ where $0<p_1<\cdots<p_n$ and $c_j$ are real numbers (not simulataneously zero), be a function on the set of positive real ...

**12**

votes

**1**answer

527 views

### Testing whether $e^x+ax^2+bx+c$ has a zero

What is the simple test with exponential polynomials to determine whether
$$f(x)=e^x+ax^2+bx+c$$ has a positive zero?
This was prompted by the question about discriminants here. We have an ineffective ...

**42**

votes

**1**answer

2k views

### Is there a nullstellensatz for trigonometric polynomials?

Let
$$ f(x) = \sum_{j=1}^n c_j e^{2\pi i\alpha_j x}, g(x) = \sum_{k=1}^m d_k e^{2\pi i\beta_k x}$$
be two (quasi-periodic) trigonometric polynomials, where the coefficients $c_j, d_k$ are complex and ...

**0**

votes

**0**answers

17 views

### Integral of the product l of the pdf of two univariate gaussian regression models with respect to the parameter

good evening to everyone. I am having trouble finding a way of calculating the integral of the product of the pdfs of two univariate Gaussian regression models with respect to the parameter, and I was ...

**1**

vote

**2**answers

144 views

### Link btw. exponential and derivatives from an algebraic perspective [closed]

I have been attempting to understand my math education (as a bachelor in electrical engineering) from a more algebraic perspective recently. I would like to understand more about the link between ...

**1**

vote

**0**answers

185 views

### Way to express a number in its most compact sum of powers

Given a non-negative number n, what is the best approach to find the most compact representation for n in terms of sums of powers, such that the bases and the exponents can't surpass a given value (...

**1**

vote

**0**answers

77 views

### Real root isolation for exponential polynomials

Suppose we are given an exponential polynomial $f:\mathbb{R}\mapsto\mathbb{R}$
$$
f(t)=\sum_{i=1}^n p_i(t)e^{\lambda_i t}
$$
where $p_i(t)$s are polynomials with algebraic coefficients and $\lambda_i$...

**3**

votes

**1**answer

164 views

### positive root for exponential polynomial

Suppose $\lambda_1,\lambda_2,\cdots,\lambda_n$ are algebraic numbers. $P_1(t),P_2(t),\cdots,P_n(t)$ are polynomials with algebraic coefficients.
The question is to whether the following question is ...

**2**

votes

**0**answers

211 views

### lower bounding the absolute value of a determinant

In a problem that I'm working on currently, the following question came up and I feel this should be fairly elementary, but I couldn't prove it myself/couldn't find a reference. Any pointers or ...

**0**

votes

**1**answer

148 views

### Express as Meijer-G function

I want to express $e^{x^2}$ as MeijerG function?
it would be possible? or what?
can i use $e^x$ MeijerG expression for this one?

**5**

votes

**1**answer

326 views

### linear independence of exponentials

Let $X$ be the set of functions $e^{p(x)}$ of the real vector $x$, where $p$ is a multivariate polynomial with $p(0)=0$.
Is any finite subset of $X$ linearly independent? If yes, why? If no, is the ...

**4**

votes

**0**answers

233 views

### Identities for powers of functions based on generalization of Lagrange interpolation

Lagrange polynomial can be used to obtain an identity:
$$(k+t)^n = \sum_{i=0}^n (k+d_i)^n \prod_{\substack{j=0\\ j\not=i}}^n \frac{t-d_j}{d_i-d_j},$$
which holds for any integer $n>0$, any real ...

**9**

votes

**1**answer

888 views

### Number of real roots of an exponential polynomial

Let $a_1, a_2, \dots, a_n$ and $b_1, b_2, \dots, b_n$ be real numbers, and assume that $\{a_i\} \neq \{b_i\}$. Can the equation
$$ e^{a_1 x} + e^{a_2 x} + \dots + e^{a_n x} = e^{b_1 x} + e^{b_2 x} + \...

**1**

vote

**3**answers

239 views

### Decidability of sum of powers exponential diophantine equation

I want to ask about decidability of exponential Diophantine equation:
$z_12^{\eta_1} + \ldots + z_n2^{\eta_n} = z$, where $z_i,\eta_i,z \in \mathbb{Z}$, and $\eta_i$ - are variables.
Can we find ...

**1**

vote

**0**answers

84 views

### Closed form solution for DDEs?

I am solving the equation $X−A−B e^{−Xy}−C e^{−X z}=0$ where $X, A, B$ and $C$ are 2x2 matrices and $y$ and $z$ are scalars. What will be the closed form solution ...

**0**

votes

**1**answer

91 views

### What is exponentially fitted osculating straight line?

While reading an article about iterative methods for solving nonlinear equations I can't understand what is exponentially fitted osculating straight line. Could someone please briefly explain this ...

**1**

vote

**0**answers

142 views

### exponential and anisotropic torus

Let $F$ be a local p-adic field and $G$ a semisimple simply connected group over $F$, $\mathfrak{g}$ its Lie algebra. Let $T$ a maximal anisotropic torus of $G$, split over an etale extension of $F$ ...

**0**

votes

**0**answers

786 views

### How to solve definite integral involving exponential function

I am trying to get a closed form for the following definite integral:
$$f(\theta)= \int_\frac{\pi}{2}^\pi \frac{1}{\sqrt{1-\alpha^2 \cos^2\theta}}\exp\left(C_2\cos\theta-C_1\sqrt{1-\alpha^2 \cos^2\...

**8**

votes

**2**answers

1k views

### Why are there so few zero-dimensional polynomial system solvers and is this because there is no real market for them?

My questions involve the quotes below from wikipedia regarding solving polynomial systems, which given the size of the market for Big Data & Predictive Analysis applications I find puzzling:
"...

**1**

vote

**0**answers

99 views

### Estimating decay of certain trigonometric polynomials

For $p=0,1,2,\dots$ and $n=0,1,2,\dots,$, let $f_{n,p}(z)=\sum_{k=0}^n k^p z^k$ be a sequence of polynomials. Restricted to the unit circle, the functions $g_{n,p}(t):=f_{n,p}(e^{it})$ are ...

**1**

vote

**0**answers

67 views

### Whether r.v. with p.g.f. $\exp [\sum\limits_{i = 1}^\infty {{q_i}({z^i}} - 1)]$ is overdispersion?

When discrete r.v. $X$ is not Poisson distributed and ${\rm{Var}}X,EX < \infty $, I want to know whether r.v. $X$ with p.g.f. $\exp [\sum\limits_{i = 1}^\infty {{q_i}({z^i}} - 1)],({q_i} \in {\rm{...

**4**

votes

**2**answers

910 views

### Efficient algorithms to determine the roots of: $p(x) = r^x $ in the finite field $GF(q)$, where $r$ a primitive root of the field

I need to make sure that no efficient (i.e., polynomial time) algorithm exists for the following problem:
Exponentiating Polynomial Root Problem (EPRP)
Let $p(x)$ be a polynomial with $\deg(p) \geq ...

**11**

votes

**2**answers

427 views

### Is equivalence of functions built from nested exponentiations a decidable problem?

Let $\mathcal{E}$ be the minimal set of symbolic expressions (without any predefined meaning) such that
The symbol $x$ is in $\mathcal{E}$, and
If expressions $P,Q\in\mathcal{E}$, then the ...

**24**

votes

**2**answers

602 views

### Order type of the smallest set containing the identity function and closed under exponentiation

Let $E$ be the smallest set of functions $\mathbb{N}^+\to\mathbb{N}^+$ containing the identity function $n \mapsto n$ and closed under exponentiation $(f,g) \mapsto \left(n \mapsto f(n)^{g(n)}\right)$...

**0**

votes

**1**answer

239 views

### Inequality of Partial Taylor Series

Hi,
For a given $\theta < 1$, and $N$ a positive integer, I am trying to find an $x > 0$ (preferably the smallest such $x$) such that the following inequality holds:
$$\sum_{k=0}^{N} \frac{x^k}...

**0**

votes

**0**answers

108 views

### maximum of the sum of polynomials

Hi
I have $n$ polynomials, each one is positive over a certain range and the maximum value each can attain is 1. Also each polynomial has atmost one peak(maximum). Is there some way that I can show ...

**0**

votes

**0**answers

258 views

### solving in x involving both exponential and logarithmic function

Is it possible to solve a function with both exponential and logarithm such as
$a x^2 - b.\log(x) = c $
in closed form; where $a,b,c$ are constants and $a>0$ and $b>0$?

**2**

votes

**3**answers

3k views

### Sum of products of exponentials and polynomials

Hi,
I am looking for a closed-form expression for the finite sum of the product of an exponential function with a polynomial function --- that is, the sum
...

**7**

votes

**2**answers

611 views

### Sturm chain analogue for exponential polynomials?

I'm going to define an exponential polynomial of degree $k$ as a function $f$ of the form
$f(x) = \sum_{i=1}^k c_ie^{\alpha_ix}$ ($\alpha_i$s real).
My first question is: is there an algorithm for ...

**7**

votes

**1**answer

486 views

### Zeros of a combination of exponentials

Is there any known result about the necessary and sufficient conditions for the existence of zeros for a function $f(x)=\sum_{n=1}^{N} a_n e^{b_n x}$, where $a_n,b_n \in \mathbb{R}\, \forall n=1,2,\...

**0**

votes

**0**answers

271 views

### Introductory text book for Linear Recurrence Sequences

What is a good introductory text for linear recurrence sequences?
What all are the necessary prerequisite for it? (My background is in Euclidean Fourier Analysis.) After browsing through several ...

**3**

votes

**1**answer

1k views

### Queries about the Skolem-Mahler-Lech theorem (integer zeros of exponential polynomials)

The Skolem-Mahler-Lech Theorem says that the integer zeros of an exponential polynomial are the union of complete arithmetic progressions and a finite number of exceptional zeros. http://terrytao....

**3**

votes

**1**answer

557 views

### Relation between entire function of exponential type and exponential polynomials

Is it true in general that the theory of entire function of exponential type and and that of exponential polynomials (with purely imaginary exponents) are analogous ?
Can one derive results about ...

**1**

vote

**3**answers

571 views

### How to find the almost period of an exponential polynomial

Let $u(t) = \Sigma_{k=1}^n c_k e^{i \lambda_k t} (c_k \in \mathbb C, \lambda_k \in \mathbb R) $ be an exponential polynomial of order $n$ with purely imaginary exponents. We can assume that the ...

**8**

votes

**4**answers

3k views

### Approximation by exponential polynomials

Let $u(t) = \Sigma_{k=1}^n c_k e^{\lambda_k t} (c_k \in \mathbb C, \lambda_k \in \mathbb C) $ be an exponential polynomial of order $n$.
Define $E_n$ to be the collection of all exponential ...

**8**

votes

**1**answer

923 views

### Salem Inequality

I have come across this inequality in the paper "Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type" http://www.math.msu.edu/~fedja/...

**5**

votes

**1**answer

1k views

### estimates of exponential polynomials

Let $ p(t) = \Sigma_{k=1}^n c_k e^{i \lambda_k t}$ be an exponential polynomial.
In the paper "Local estimates for exponential polynomials and their applications to inequalities of the uncertainty ...

**7**

votes

**3**answers

2k views

### Rational exponential expressions

Consider the following extension of polynomials. The rational exponential expressions (REXes) are given by:
The leaves 1 and $x$ for $x$ drawn from a class of variables; and
Closed under the binary ...