Questions tagged [exponential-polynomials]
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43 questions
45
votes
1
answer
3k
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 ...
- 114k
23
votes
2
answers
669
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 $(f,g) \mapsto \left(n \mapsto f(n)^{g(n)}\right)$...
- 872
12
votes
1
answer
1k
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 ...
user44143
11
votes
1
answer
1k
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} + \...
- 1,474
10
votes
2
answers
455
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 ...
- 1,699
9
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:
"...
- 159
8
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 ...
- 2,283
8
votes
4
answers
4k
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 ...
- 1,795
8
votes
1
answer
1k
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/...
- 1,795
8
votes
1
answer
591
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,\...
- 83
7
votes
2
answers
724
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 ...
- 8,688
5
votes
1
answer
728
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 ...
- 3,873
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 ...
- 1,795
4
votes
2
answers
1k
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 ...
4
votes
0
answers
302
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},$$
which holds for any integer $n>0$, any real ...
- 34.3k
3
votes
1
answer
192
views
positive root for exponential polynomial
Suppose $\lambda_1,\lambda_2,\cdots,\lambda_n$ are algebraic numbers. $P_1(t),P_2(t),\cdots,P_n(t)$ are polynomials with algebraic coefficients.
The question is to whether the following question is ...
- 1,503
3
votes
1
answer
120
views
How to establish regions of convexity/concavity of a ratio of exponential polynomials?
Problem:
Let $f\colon \mathopen[0,1\mathclose] \to \mathbb{R}$ be defined as
$$
f(x) = \frac{e^{\rho x}-1}{e^{\rho x}-1+e^{\rho (1-\gamma x)}-e^{\rho (1-\gamma) x}}
$$
where $\rho$ and $\gamma$ are ...
- 33
3
votes
1
answer
2k
views
Queries about the Skolem-Mahler-Lech theorem (integer zeros of exponential polynomials)
The Skolem-Mahler-Lech Theorem says that the integer zeros of an exponential polynomial are the union of complete arithmetic progressions and a finite number of exceptional zeros. http://terrytao....
- 1,795
3
votes
1
answer
675
views
Relation between entire function of exponential type and exponential polynomials
Is it true in general that the theory of entire function of exponential type and and that of exponential polynomials (with purely imaginary exponents) are analogous ?
Can one derive results about ...
- 1,795
2
votes
3
answers
5k
views
Sum of products of exponentials and polynomials
Hi,
I am looking for a closed-form expression for the finite sum of the product of an exponential function with a polynomial function --- that is, the sum
...
2
votes
3
answers
632
views
How to find the almost period of an exponential polynomial
Let $u(t) = \Sigma_{k=1}^n c_k e^{i \lambda_k t} (c_k \in \mathbb C, \lambda_k \in \mathbb R) $ be an exponential polynomial of order $n$ with purely imaginary exponents. We can assume that the ...
- 1,795
2
votes
0
answers
108
views
Absolute lower bound on derivative of generalized trigonometric polynomial at zeroes
By a generalized trigonometric polynomial, I shall mean a function $f:\mathbb R^+ \rightarrow \mathbb R$ given by an expression of the form
$$f(x) := \sum_{j=1}^k a_j \cos(\alpha_j x) + b_j \sin(\...
- 739
2
votes
0
answers
402
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 ...
- 305
1
vote
3
answers
284
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 ...
- 11
1
vote
1
answer
90
views
The number of roots of pseudo-exponential polynomials
Assume that $J$ is the interval $(-\pi,\pi]$. For $k=1,\ldots,2n$, suppose that $\lambda_k$s are real functions on $J$ with $|\lambda_k|=1$, meaning that $\lambda_k(t)$ is either $-1$ or $1$ where $t\...
- 4,058
1
vote
1
answer
118
views
A sine type Chebyshev system
A sequence of real functions $\{\phi_1,\cdots,\phi_n\}$ is called a Chebyshev system on an interval $I\subseteq\mathbb{R}$, if any real linear combination $\sum_{l=1}^n a_l\phi_l$ has at most $n-1$ ...
- 4,058
1
vote
0
answers
74
views
Decidability of a polynomial-exponential equation in two variables
My question is with regards to the following (algorithmic) problem:
Problem. Given $f\in \mathbb{Z}[x,y], a,b\in \mathbb{Q}, r\in \mathbb{Z}$, do there exist positive integers $m,n$ such that $f(m,n) =...
- 151
1
vote
0
answers
128
views
Self-referencing recurrence relation with exponential
I have the self-referencing recurrence relation
$$
d(0) = 0
$$
$$
d(1) = a
$$
$$
d(n+1) = d(n) + a*e^{-(\frac{d(n)-n*c}{b})^4}
$$
Written as a sum:
$$
d(N) = a*\sum_{n=0}^{N}e^{-(\frac{d(n)-n*c}{b})^...
- 11
1
vote
2
answers
157
views
Link btw. exponential and derivatives from an algebraic perspective [closed]
I have been attempting to understand my math education (as a bachelor in electrical engineering) from a more algebraic perspective recently. I would like to understand more about the link between ...
1
vote
0
answers
222
views
Way to express a number in its most compact sum of powers
Given a non-negative number n, what is the best approach to find the most compact representation for n in terms of sums of powers, such that the bases and the exponents can't surpass a given value (...
- 11
1
vote
0
answers
102
views
Real root isolation for exponential polynomials
Suppose we are given an exponential polynomial $f:\mathbb{R}\mapsto\mathbb{R}$
$$
f(t)=\sum_{i=1}^n p_i(t)e^{\lambda_i t}
$$
where $p_i(t)$s are polynomials with algebraic coefficients and $\lambda_i$...
- 1,503
1
vote
0
answers
90
views
Closed form solution for DDEs?
I am solving the equation $X−A−B e^{−Xy}−C e^{−X z}=0$ where $X, A, B$ and $C$ are 2x2 matrices and $y$ and $z$ are scalars. What will be the closed form solution ...
- 111
1
vote
0
answers
197
views
exponential and anisotropic torus
Let $F$ be a local p-adic field and $G$ a semisimple simply connected group over $F$, $\mathfrak{g}$ its Lie algebra. Let $T$ a maximal anisotropic torus of $G$, split over an etale extension of $F$ ...
- 3,472
1
vote
0
answers
104
views
Estimating decay of certain trigonometric polynomials
For $p=0,1,2,\dots$ and $n=0,1,2,\dots,$, let $f_{n,p}(z)=\sum_{k=0}^n k^p z^k$ be a sequence of polynomials. Restricted to the unit circle, the functions $g_{n,p}(t):=f_{n,p}(e^{it})$ are ...
- 959
1
vote
0
answers
69
views
Whether r.v. with p.g.f. $\exp [\sum\limits_{i = 1}^\infty {{q_i}({z^i}} - 1)]$ is overdispersion?
When discrete r.v. $X$ is not Poisson distributed and ${\rm{Var}}X,EX < \infty $, I want to know whether r.v. $X$ with p.g.f. $\exp [\sum\limits_{i = 1}^\infty {{q_i}({z^i}} - 1)],({q_i} \in {\rm{...
- 113
0
votes
1
answer
98
views
What is exponentially fitted osculating straight line?
While reading an article about iterative methods for solving nonlinear equations I can't understand what is exponentially fitted osculating straight line. Could someone please briefly explain this ...
- 163
0
votes
1
answer
235
views
Express as Meijer-G function
I want to express $e^{x^2}$ as MeijerG function?
it would be possible? or what?
can i use $e^x$ MeijerG expression for this one?
0
votes
1
answer
290
views
Inequality of Partial Taylor Series
Hi,
For a given $\theta < 1$, and $N$ a positive integer, I am trying to find an $x > 0$ (preferably the smallest such $x$) such that the following inequality holds:
$$\sum_{k=0}^{N} \frac{x^k}...
- 51
0
votes
1
answer
346
views
Solving an equation involving $x$ both squared and inside a logarithm [closed]
Is it possible to solve a function with both exponential and logarithm such as
$$
a x^2 - b\cdot\log(x) = c
$$
in closed form; where $a,b,c$ are constants and $a>0$ and $b>0$?
- 109
0
votes
1
answer
151
views
Is it possible to solve for y in this equation? [closed]
Is it possible to solve for $y$ in this equation? $$-y^{-x}+y-1=0$$ People have mentioned the use of the Lambert W function or other non-elementary functions, but I haven't been able to make use of ...
0
votes
0
answers
804
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\...
- 1
0
votes
0
answers
130
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 ...
- 1
0
votes
0
answers
325
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 ...