# Questions tagged [exponential-polynomials]

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0answers
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### 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 ...
1answer
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 ...
1answer
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 ...
0answers
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 ...
2answers
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 ...
0answers
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 (...
0answers
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 $\lambda_i$...
1answer
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 ...
0answers
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 ...
1answer
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?
1answer
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 ...
0answers
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 ...
1answer
888 views

2answers
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: "...
0answers
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 ...
0answers
67 views

2answers
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 ...
2answers
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 $(f,g) \mapsto \left(n \mapsto f(n)^{g(n)}\right)$...
1answer
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}...
0answers
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 ...
0answers
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 $b>0$?
3answers
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 ...
2answers
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 ...
1answer
486 views