Questions tagged [exponential-polynomials]

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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) =...
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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})^...
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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(\...
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12 votes
1 answer
557 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 ...
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45 votes
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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 ...
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2 answers
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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 ...
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1 vote
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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 (...
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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 $\lambda_i$...
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3 votes
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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 ...
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2 votes
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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 ...
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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?
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5 votes
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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 ...
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4 votes
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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 ...
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9 votes
1 answer
997 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} + \...
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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 ...
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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 ...
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1 answer
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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 ...
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1 vote
0 answers
170 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 $F$ ...
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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\...
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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: "...
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1 vote
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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 ...
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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{...
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4 votes
2 answers
946 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 ...
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11 votes
2 answers
434 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 ...
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24 votes
2 answers
614 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 $(f,g) \mapsto \left(n \mapsto f(n)^{g(n)}\right)$...
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1 answer
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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}...
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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 ...
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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 $b>0$?
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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 ...
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7 votes
2 answers
629 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 ...
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7 votes
1 answer
510 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,\...
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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 ...
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1 answer
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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....
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3 votes
1 answer
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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 ...
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1 vote
3 answers
589 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 ...
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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 ...
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8 votes
1 answer
974 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/...
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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 ...
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7 votes
3 answers
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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 ...
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