A conjectural formula for the "minimal degree function", $k\rightarrow d\min(k)$, attached to recursion, $f\rightarrow A(f)$, in char $3$

Question

THE RECURSION: $f\rightarrow A(f)$

$A: t(\mathbb{Z}/3)[t^3]\rightarrow t(\mathbb{Z}/3)[t^3]$ is the $\mathbb{Z}/3$-linear map with $A(t)=0, A(t^4)=t, A(t^7)=t^4$, and $A((t^9)f)=(2t^9)A(f)+(t^3)A((t^3)f)$.

An induction shows that $A(t^k)$ is either $t^{k-3} + O(t^{k-6})$ or $2t^{k-3} + O(t^{k-6})$ when $k$ is $4$ or $7 \bmod 9$, and is $O(t^{k-6})$ when $k$ is $1 \bmod 9$. (Here $O(t^m)$ is shorthand for an element of degree at most $m$ in $t$)

$\mathbf{d\min (k)}$

Let $N'$ consist of all positive integers that are $1\bmod 3$. For $k$ in $N'$, $d\min(k)$ is the smallest degree of $A(f)$, $f$ running over the elements of $t(\mathbb{Z}/3)[t^3]$ of exact degree $k$. Note that $d \min$ is a function from $N'$ to the union of $N'$ with $-\infty$. By the paragraph above $d\min (k)$ is $k-3$ when $k$ is $4$ or $7\bmod 9$. Suppose however that $k$ is $1 \bmod 9$; the last sentence shows that $d\min(k)$ cannot be $1$ or $4\bmod 9$ and so must be $7\bmod 9$. Also by the last paragraph $d\min(k)$ is at most $k-6$; we conclude that $d\min(k)$ is at most $k-12$.

EXAMPLES

$A(t^{10} - t^{7})=0$, and $d\min(10)=-\infty$

$A(t^{19} + t^{16} + t^{13})=2t^{7}$, and $d\min(19)=7$.

SOME RESULTS FOR $d \min(k)$

We've seen that when $k$ is $4$ or $7\bmod 9$, $d\min(k)=k-3$. More labor shows:

(a) When $k$ is $19\bmod 27$, $d\min(k)=k-12$

(b) When $k$ is $37$ or $64\bmod 81$, $d\min(k)=k-21$

(c) When $k$ is $55\bmod 81$, $d\min(k)=k-30$.

The question remains--what is $d\min(k)$ for $k$ in the three remaining congruence classes $\bmod 81$--the classes of $1,10,$ and $28$. (These are the only classes containing $k$ with $d\min(k)=-\infty$). We now present a conjectural answer.

THE QUESTION

Do the following hold?

1. $d\min(81n+1)=9d\min(9n+1) -11$

2. $d\min(81n+10)=9d\min(9n+1) +7$

3. $d\min(81n+28)=9d\min(9n+1) +16$

Tim Hickey has kindly provided me with a computer program that establishes that 1,2, and 3 hold for $n$ up to $100$.

MOTIVATION

An elementary proof of 1,2 and 3 should lead an alternative proof of the Bellaiche-Khare-Medvedovsky result about the structure of the Hecke algebra attached to the space of $\bmod 3$ elliptic modular forms of level 1. And there are similar empirically verified conjectures for (much!) more complicated recursions whose proof would lead to an understanding of the structure of related Hecke algebras in levels $\Gamma_0(5)$ and $\Gamma_0(13)$.