2

I was wondering what are some open problems (even if deemed impossible) in basic function theory (stuff you'd learn in high school) and/or open problems to do with polynomials...

Thank you in advance :)

Brandon Battye
  • 115
  • 1
    In the case of polynomials (in particular if all coefficients are rational), I can barely imgaine an open or undecidable problem. If functions like $\sin(x)$ are also allowed, the integration can be problematic, but the Risch-algorithm can decide in almost every case whether an antiderivate exists or not. Really pathological cases are usually not discussed in high school, so for the problems high school studens have to deal with, there will usually exist an algorithm to solve the execise. – Peter May 15 '20 at 09:50
  • 2
    The professor of the answer (below) also refered in the past a survey (from his answer for Question 1) for the known as Casas-Alvero conjecture, from the post Casas-Alvero conjecture: difficulty and analogous conjecture for integers of this Mathematics Stack Exchange (Dec 28 '17 ), question with identificator 2582947 in this site. Wikipedia has an article for this conjecture with title Casas-Alvero conjecture – user759001 May 15 '20 at 09:51
  • 2
    The link is here on MSE. – Dietrich Burde May 15 '20 at 10:02

1 Answers1

2

There is the so called Bunyakovsky conjecture for polynomials $f(x)\in \Bbb Z[x]$. It says that $f$ has infinitely many prime values in the sequence $f(1),f(2),f(3),\cdots$, provided $f$ has a positive leading coefficient, $f$ is irreducible in $\Bbb Z[x]$ and the above values are coprime.

For example, it is conjectured that $f(x)=x^2+1$ produces infinitely many primes.

References:

Primes of the form $n^2+1$ - hard?

Dietrich Burde
  • 130,978
  • I think the question is more about analysis. Of course, in number theory, there are many open problems , also in connection with polynomials. – Peter May 15 '20 at 09:53
  • @Peter In the first part of the question, I agree. But then he says "open problems to do with polynomials...even if deemed impossible". A perfect match. The solution may require a lot of analysis, by the way. But you are right, the question probably is not meant so broad. – Dietrich Burde May 15 '20 at 09:55
  • From this, I looked at Landau's problems... Legendre's conjecture looks really bloody interesting! – Brandon Battye May 15 '20 at 10:26