During some casual investigation of polynomials over an integer ring $\mathbb{Z}_n$ (or $\mathbb{Z}/n\mathbb{Z}$ if you prefer), I noticed that some polynomials induce the same map. I'm curious about how one could tell if two polynomials are equivalent in this way without checking directly.

For example, in $\mathbb{Z}_8$, $f(x) = 2x^3 + 5x + 3$ is the same as $g(x)=x^4 + 3x^2 + 3x + 3$. They are bijective and act on $\mathbb{Z}_8$ as the permutation $(3, 2, 5, 0, 7, 6, 1, 4)$.

My main question is: What are the criteria for two polynomials to induce the same map on $\mathbb{Z}_n$?

I'm also interested in other information about this, such as: For a given polynomial, are there infinitely many equivalent polynomials? Will the lowest-degree polynomial in an equivalence class always have a degree less than $n$? Does it matter what kind of number $n$ is (e.g. prime or composite)? Are there unique polynomials with $deg\geq1$ having no equivalent?

Without knowing much about this situation, my guess is that the Chinese Remainder Theorem, Euler's Theorem, and/or Fermat's Little Theorem will come into play. I'm exploring a bit outside of my mathematical comfort zone and I have very little experience with number theory, so this is where I get kind of lost.

  • $\begingroup$ Euler, and Polynomial remainder theorem may be of use. $\endgroup$ – Roddy MacPhee Oct 9 at 14:57
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    $\begingroup$ Try looking at $g-f$. This is a polynomial that is zero for any value of $x$. In this case it factorises very nicely as a product of four consecutive numbers - these will always have a multiple of 4 and another even number among them, so the product is 0 mod 8. For a prime modulus $p$, you have $x^p-x=0$ for all $x$, so you can add any multiple of this to a polynomial to get an equivalent one (and one of these will have degree less than $p$). I'm not sure what you'd need for composite moduli in general (at least not if it has to work even when $x$ and $n$ are not coprime). $\endgroup$ – Jaap Scherphuis Oct 9 at 15:36
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    $\begingroup$ See also math.stackexchange.com/questions/3387540/… $\endgroup$ – lhf Oct 9 at 23:47

Two polynomials induce the same function iff their difference induces the zero function.

Here is a general result about polynomials that induce the zero function:

If $r$ is the maximum exponent in the prime factorization of $n$, then $x \mapsto x^{r+\lambda (n)}-x^r$ is the zero function mod $n$. [Wikipedia]

Here, $\lambda$ is the Carmichael function.

I don't know whether this is the polynomial of least degree that induces the zero function mod $n$.

  • $\begingroup$ Can you expand on how this relates to polynomials with more terms and coefficients? If I'm not mistaken, $f-g$ from my example is $x^4+6x^3+3x+6x$ and the $\lambda$ version of the zero function is $x^5+7x^3$. $\endgroup$ – Era Oct 9 at 17:49
  • $\begingroup$ @Era, you're quite right. So, in this case, $x^5+7x^3$ is not the polynomial of least degree that induces the zero function mod $8$. $\endgroup$ – lhf Oct 9 at 18:17
  • $\begingroup$ The polynomial of least degree that induces the zero function mod $8$ is $4x^2+4x$. $\endgroup$ – lhf Oct 9 at 18:27
  • $\begingroup$ The monic polynomial of least degree that induces the zero function mod $8$ is $x^4+2x^3+3x^2+2x$, which is your $f-g$, up to sign. $\endgroup$ – lhf Oct 9 at 18:34
  • $\begingroup$ The quoted theorem is a special case of the theorem I prove here. and is also proved elsewhere here. $\endgroup$ – Bill Dubuque Oct 9 at 19:09

By Euler's totient function, $$x^{\varphi (n)\cdot y+k}\equiv x^k\bmod n$$

Meaning any univariate polynomial, with degree more than $$\varphi(n)$$ is redundant as the coefficients from higher terms is congruent to adding it to the coefficient of a lower degree term.

Furthermore, by polynomial remainder theorem, the values of the polynomial differing in input by $n$ are also congruent.

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    $\begingroup$ Can you explain what you mean by "by Euler's totient function"? I know what the function is, but how does it produce this result? Is that a theorem? $\endgroup$ – Era Oct 9 at 16:54
  • $\begingroup$ it's an extension of the normal use, just like my extension of Fermat, used to find the remainder of a 78 digit number on division by 7. $\endgroup$ – Roddy MacPhee Oct 9 at 16:56
  • $\begingroup$ math.stackexchange.com/questions/3152587/… $\endgroup$ – Roddy MacPhee Oct 9 at 16:57
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    $\begingroup$ You're making an obvious mistake (what if $n=4$ and $x=2$?). $\endgroup$ – metamorphy Oct 9 at 17:09
  • $\begingroup$ then only the last 2 terms matter... $\endgroup$ – Roddy MacPhee Oct 9 at 17:26

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