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Solutions of polynomial equations in several variables modulo a prime power

Arnaud Bodin, Christian Drouin

Abstract

We explain how to obtain the set of solutions of a multivariate polynomial equation modulo a power of a prime number. These solutions are determined by a tree, called the trunk, which makes it possible to reconstruct all solutions. We apply these methods to determine the number of solutions, without having to enumerate them. We also illustrate these techniques by proving a simple case of Igusa's theorem: the Poincaré series associated with a polynomial in two separated variables is rational.

Solutions of polynomial equations in several variables modulo a prime power

Abstract

We explain how to obtain the set of solutions of a multivariate polynomial equation modulo a power of a prime number. These solutions are determined by a tree, called the trunk, which makes it possible to reconstruct all solutions. We apply these methods to determine the number of solutions, without having to enumerate them. We also illustrate these techniques by proving a simple case of Igusa's theorem: the Poincaré series associated with a polynomial in two separated variables is rational.
Paper Structure (42 sections, 19 theorems, 106 equations, 4 figures)

This paper contains 42 sections, 19 theorems, 106 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Objectives
  3. Solutions modulo $p^e$
  4. The Poincaré series
  5. The trunk
  6. Thickness
  7. Solution tree
  8. Trunk
  9. Tree-top function
  10. From the trunk to the tree
  11. Properties of the thickness
  12. Computation of the thickness
  13. The thickness decreases along the trunk
  14. Hensel's lemma
  15. The number of solutions
  16. ...and 27 more sections

Key Result

Theorem 1.1

Let $P \in \mathbb{Z}[x_1,\ldots,x_n]$. The number $N_e$ of solutions of the equation $P(x_1,\ldots,x_n) \equiv 0 \pmod{p^e}$ in $(\mathbb{Z}/ p^e \mathbb{Z})^n$ is:

Figures (4)

  • Figure 1: A Hensel tree for $n=2$ variables, drawn here for $p=3$. From each vertex there are $p$ outgoing branches. All vertices have thickness $1$.
  • Figure 2: The trunk up to height $3$. The drawing is for $p=3$. The dotted parts and the numbers of branches correspond to the cases $p>3$.
  • Figure 3: A branch attached to a vertex of odd height $2l+1$ of the principal stalk. There are $p-1$ such outgoing branches from this vertex. The branches are finite and stop at height $3l+2$. All vertices have thickness $2$, except the terminal vertices, which have thickness $1$.
  • Figure 4: Two branches attached to a vertex of even height $2l$ (with $l>0$) of the principal stalk. On the right, there are $\frac{p-1}{2}$ finite outgoing branches up to height $3l$, corresponding to integers that are not squares modulo $p$; all their vertices have thickness $2$. On the left, there are also $\frac{p-1}{2}$ outgoing branches corresponding to nonzero integers that are squares modulo $p$; the vertices up to height $3l$ have thickness $2$, and on each of the $p^{l-1}$ vertices at height $3l$ are attached $2p$ Hensel trees.

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • proof
  • ...and 32 more