All Questions
Tagged with exponential-polynomials nt.number-theory
7 questions
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
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
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
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
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 ...
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
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