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Let $q=2^h$ and $t=2^r$ for some $h\ge r$ and denote by $\mathbb{F}_q$ the finite field of order $q$.

(since the previous, simple version was wrong, I'm posting here a new version)

Let $f$ be a monic polynomial with $f(0)=0$, $f(1)=0$, such that the equation $f(z)=c$ has either $0$ or $t$ solutions in $\mathbb{F}_q$. Let $F_a(z)=\frac{f(z)+f(a)}{z+a}$, and we are given that the restriction of $F_a(z)$ to its non-roots (and $\neq a$) is injective, and there are constants $w_1,...,w_{t-1}$ such that none of the polynomials $w_i^{-1}+F_a(z)$ has a root in $\mathbb{F}_q\setminus\{a\}$.

Computationally, for $q\le 32$, I turns out that $\{0,w_1,\ldots,w_{t-1}\}$ is always an additive subgroup in $\mathbb{F}_q$, and that the $\{z|f(z)=c\}$ are also cosets of an additive subgroup. I wonder if any of this could be proven in general.

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    $\begingroup$ This seems interesting. What is the motivation? $\endgroup$ – Slade Oct 16 '14 at 12:51
  • $\begingroup$ Are the $w_i$ supposed to be distinct, or correspond to $a$ in some way? $\endgroup$ – Slade Oct 16 '14 at 13:39
  • $\begingroup$ The $\{w_i | 1\le i\le t-1\}$ are a fixed set for which a polynomial $f$ with said properties exist. If it helps, it can be derived from the above properties that when writing $f(z)=\sum_{i=0}^{q-1} a_i z^i$ then $a_1=\sum_{i=1}^{t-1} w_i^{-1}$ and that $a_{2s+1}=0$ for all $s\ge 1$, and also that $a_{q-2}=0$. $\endgroup$ – user1111929 Oct 16 '14 at 13:53
  • $\begingroup$ It does look like you are seeing linearized polynomials. I don't see right away why that should be the case though. $\endgroup$ – Jyrki Lahtonen Oct 17 '14 at 18:02
  • $\begingroup$ The motivation by the way is proving a property of certain point sets in projective geometry over finite fields. A proof of this (even just the {0,w1,...,wt} part) can be completed by me to a good research article (web of science level). I will gladly offer coauthorship to whoever provides me with a proof. $\endgroup$ – user1111929 Oct 17 '14 at 19:43
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If $t\neq 1,2,q$, then there exist subsets of $\mathbb{F}_q$ of size $t$ that include $0$, but that are not additive subgroups. For example, take $S = (\mathbb{F}_t\setminus \{1\}) \cup \{a\}$, where $a\notin \mathbb{F}_t$.

Since any function from $\mathbb{F}_q$ to itself can be interpolated with a polynomial, this is quite false as stated—just let $f$ be any $t$-to-$1$ function with $f(S) = 0$—though I'm still curious what the idea is, and whether there is a less trivial statement of the problem.

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  • $\begingroup$ You are right, I have probably cut out too much of the original problem. I have updated it to the full problem statement now. $\endgroup$ – user1111929 Oct 16 '14 at 13:23
  • $\begingroup$ I updated the problem statement even more. These $q/t+1$ sets all appear to be cosets of additive subgroups of size $t$ of $\mathbb{F}_q$. $\endgroup$ – user1111929 Oct 16 '14 at 13:33

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