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I found this problem in a number theory course, I am assuming (but not sure) it is supposed to be an application of Hensel's lemma.

For every $n \in \mathbb{N}_0$, determine the number of solutions over $\mathbb{Z}/5^n\mathbb{Z}$ of the equation $y^2 = x^3 + x^2 + 5$.

In the base case $n = 1$, there are four solutions $(0, 0), (3, 1), (3, 4), (4, 0)$. The problem with Hensel's lemma in the multivariate case seems to be, if you lift your root to a higher prime power, you can no longer be sure this new root is unique. So instead I figured for every root, I'd fix each variable separately, and consider the univariate problem in the other respective variable. From this I derived the following results:

  • $(0, 0)$ doesn't have higher power solutions by fixing either variable.
  • $(3, 1)$ has a unique higher power solution by fixing either variable.
  • $(3, 4)$ has a unique higher power solution by fixing either variable.
  • $(4, 0)$ has a unique higher power solution by fixing the second variable.

However I have no clue how to further deduce the amount of solutions apart from these implicit ones. Just to get an idea of where I was headed I computed all solutions in the case $n = 2$. This way I did notice there are patterns in the solutions:

  • $(0, 0)$ mod $5$, there are no corresponding solutions mod $25$.
  • $(3, 1)$ mod $5$, the solutions are $(3 + 5k, 21 - 5k)$ mod $25$ for each $k$.
  • $(3, 4)$ mod $5$, the solutions are $(3 + 5k, 4 + 5k)$ mod $25$ for each $k$.
  • $(4, 0)$ mod $5$, the solutions are $(19, 5k)$ mod $25$ for each $k$.

I can sense there must be a connection between the results I got in the univariate case, and the results I have to get in the multivariate case. But I can't seem to figure out how to formally argue, based on the results of Hensel's lemma, why those patterns in the solutions of $n = 2$ turned out the way they are. Neither do I have a clue how to generalize my results towards an arbitrary power of $5$.

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    $\begingroup$ Let $a \in \Bbb{Z}$ and $f(x,y) = y^2-x^3-x^2-5$. Once $g(y)=f(3+5a,y)$ splits and is separable modulo $5$ (ie. $g(1)\equiv g(-1) \equiv 0 \bmod 0, g'(1),g'(-1) \not \equiv 0 \bmod 5$) then for each $k$ there is exactly $\deg(g)=2$ solutions $y \bmod 5^k$ such that $f(3+5a,y)\equiv 0 \bmod 5^k$. $\endgroup$
    – reuns
    May 19, 2019 at 20:52

2 Answers 2

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COMMENT.- In general, every solution $x_0$ of $f (x) = 0 \mod p$, except when $\color{red}{f '(x_0) \ne 0 \mod p}$, gives a solution of $f (x) = 0 \mod p ^ n$solving successively the equation modulo $p^2,p^3,\cdots p^n$.

Because of $f (x_0) = 0\mod p ^ n\Rightarrow f (x_0) = 0 \mod p$ putting $x=x_0+px_1$ we can solve $$f(x_0+px_1)=0\mod p^2$$ getting after simplifications$$f(x_0)+px_1f'(x_0)=0\mod p^2$$ which implies $$f(x_0)+px_1f'(x_0)=0\mod p$$ so we get $$x_1=\frac{-f(x_0)}{pf'(x_0)}\mod p$$ For example the given solution $(3,1)$ modulo $5$ of $y^2=x^3+x^2+5$ gives the univariable equation $$1=x^3+x^2+5$$ which is the equation $$f(x)=x^3+x^2+4=0$$ and with the exposed methode one get $x_1=\dfrac{-f(3)}{5f'(3)}=\dfrac{-(27+9+4)}{5(3\cdot3^2+2\cdot3)}=-\dfrac 33=4$.

Thus $(3,4)$ is a solution of $y^2=x^3+x^2+5$ modulo $25$.

In the proposed equation we have to go first to $3$ distinct equations because if $a$ is a square modulo $25$ we must have $f(x)=x^3+x^2+5-a=0\mod 5$ and because of $x^3+x^2\in\{1,0,2\}$ one has $$\begin{cases}f_1(x)=x^3+x^2+5=0\\f_2(x)=x^3+x^2+4=0\\f_3(x)=x^3+x^2+3=0\end{cases}$$ Some work gives then the following solutions modulo 25: $(3,\pm4),(8,\pm9),(13,\pm11),(18,\pm6),(19,0),(23,\pm1)$.

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I think I figured it out, but I also feel like I'm trying to re-prove results that should probably follow straight out of Hensel's lemma. For completeness' sake, I'll post the version of Hensel's lemma I have here.

Let $p$ be prime, $k \in \mathbb{Z}_{>0}$, $f \in \mathbb{Z}_p[x_1, \dots, x_k], a = (a_1, \dots, a_k) \in \mathbb{Z}_p^k$ en $e \in \mathbb{Z}_{\ge 0}$. Assume

\begin{equation} f(a) \equiv 0 \bmod p^{2e + 1}, \quad \min_{1 \le i \le k} \text{ord}_p\left(\frac{\partial f}{\partial x_i}(a)\right) = e. \end{equation} Then some $b \in \mathbb{Z}_p^k$ exists such that $f(b) = 0$ and $b \equiv a \bmod p^{e+1}$.

And this is the solution I came up with.

First, consider the solutions for $k = 1$. Since $y^2$ is a square, it follows that $y \in \{0, 1, 4\}$. There are exactly four solutions $(x, y) \in (\mathbb{Z}/{5\mathbb{Z}})^2$ which are given by $z_1 = (0, 0), z_2 = (4, 0), z_3 = (3, 1), z_4 = (3, 4)$. Now consider the preconditions of the multivariate lemma of Hensel-Rychlik on the polynomial given by $f(x, y) = x^3 + x^2 + 5 - y^2$.

\begin{array}{c|c|c|c|c|c} z & (x, y) & (0, 0) & (4, 0) & (3, 1) & (3, 4) \\ \hline \frac{\delta f}{\delta x}(z) & x(3x + 2) & 0 & 1 & 3 & 3 \\ \frac{\delta f}{\delta y}(z) & -2y & 0 & 0 & 3 & 2 \\ \hline \text{min ord} & - & \infty & 0 & 0 & 0 \end{array}

Every zero except the first satisfies the preconditions, which implies at least one solution exists congruent with every zero, except (not necessarily) for $(0, 0)$. Now assume a zero $(5u, 5v) \in \mathbb{Z}/{5\mathbb{Z}}$ of the polynomial exists. Then it follows that

\begin{align*} f(5u, 5v) &\equiv (5u)^3 + (5u)^2 + 5 - (5v)^2 \bmod 25 \\ &\equiv 5 \bmod 25 \equiv 0 \bmod 25 \end{align*}

This leads to a contradiction, which means a solution congruent to $(0, 0)$ is indeed impossible. Now assume some arbitrary $a, b \in \mathbb{Z}/{5^{k + 1}\mathbb{Z}}$ for which holds that $f(a, b) \equiv 0 \bmod 5^{k + 1}$. In other words, it is congruent to one of the previously found roots. Then for some $u, v \in \mathbb{Z}/{5^{k + 1}\mathbb{Z}}$ it follows that

\begin{align*} f(a + 5^ku, b + 5^kv) &\equiv (a + 5^ku)^3 + (a + 5^kv)^2 + 5 - (b + 5^kv)^2 \bmod 5^{n + 1} \\ &\equiv a^3 + 3 \cdot a^25^ku + a^2 + 2 \cdot a5^ku + 5 - b^2 - 2 \cdot b5^kv \bmod 5^{n + 1} \\ &\equiv 3 \cdot a^25^ku + 2 \cdot a5^ku - 2 \cdot b5^kv \bmod 5^{n + 1} \\ &\equiv 5^k(3 \cdot a^2u + 2 \cdot au - 2 \cdot bv) \bmod 5^{n + 1} \\ &\equiv 0 \bmod 5^{n + 1} \\ &\,\Updownarrow \\ (3a^2 + 2a)u &\equiv 2bv \bmod 5 \end{align*}

In other words, the existence of more solutions in $\mathbb{Z}/{5^{k + 1}\mathbb{Z}}$ completely depends on their behavior in $\mathbb{Z}/{5^k\mathbb{Z}}$, which can be checked using the original three roots.

  • (a, b) = (4, 0): This implies $u \equiv 0 \bmod 5$, which means 5 solutions $(u, v)$ exist.
  • (a, b) = (3, 1): This implies $3u \equiv 2v \bmod 5$, which means 5 solutions $(u, v)$ exist.
  • (a, b) = (3, 4): This implies $3u \equiv 3v \bmod 5$, which means 5 solutions $(u, v)$ exist.

This implies, for every solution of the equation in $\mathbb{Z}/{5^k\mathbb{Z}}$, five congruent solutions exist in $\mathbb{Z}/{5^{k + 1}\mathbb{Z}}$. It can be concluded that, for $k > 1$, the amount of solutions to the equation is given by $3 \cdot 5^k$.

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