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Integral quadratic forms over a ring of $p$ -adic integers

This paper establishes necessary and sufficient conditions for diagonal integral quadratic forms to be universal over the matrix ring $M_2(\mathbb{Z}_p)$ for all primes $p$ , and subsequently derives bounds for the minimal number of variables required to ensure universality over $M_n(\mathbb{Z}_p)$ for $n \geq 3$ when at least three coefficients are $p$ -adic units.

Original authors: Mrunal Hardikar, Anuradha S. Garge

Published 2026-08-26

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CC BY 4.0

Original authors: Mrunal Hardikar, Anuradha S. Garge

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of mathematics, there is a branch dedicated to understanding how numbers can be combined to build other numbers. Imagine a set of building blocks, each with a specific weight. The question mathematicians ask is whether you can arrange these blocks to construct any possible structure you desire. This field, known as the study of quadratic forms, deals with expressions where numbers are squared and added together. For centuries, scholars have explored when such expressions can represent every single integer, a property called universality. Recently, the focus has shifted to a more complex setting involving matrices, which are grids of numbers used to describe transformations in space. Researchers are now asking if these grid-based expressions can build every possible matrix within a specific system of numbers known as p-adic integers. These numbers are a specialized extension of the integers we use every day, designed to handle infinite precision in a way that reveals deep structural patterns. Understanding whether these forms are universal helps mathematicians map the fundamental limits of how numbers interact in these intricate systems.

A team of researchers has recently provided significant bounds on the conditions required for this question for a specific type of matrix grid. They investigated whether a sum of squared terms, each multiplied by a coefficient, could generate every possible two-by-two grid of these special numbers. Their work confirms that for these grids to be capable of building any shape, the coefficients used in the sum must include at least two numbers that are "units." In this context, a unit is a number that is not divisible by the system's base, essentially acting as a versatile building block. The researchers proved that if you have at least two of these versatile blocks, you can construct any two-by-two grid. Conversely, if you have fewer than two, there will be certain grids that remain impossible to build. This finding holds true whether the system is based on the number two or any other odd prime number.

The study then expanded its scope to larger grids, specifically those with three rows and three columns. Here, the researchers sought to determine the minimum number of squared terms needed to guarantee that any three-by-three grid could be constructed. They discovered that if the system is based on an odd prime number, only three such terms are required, provided that at least three of the coefficients are versatile units. This means that with just three carefully chosen components, one can assemble any possible three-by-three matrix in these systems. However, the rules change slightly when the system is based on the number two. In this specific case, the researchers found that four terms are necessary to ensure that every possible grid can be formed. This distinction highlights how the underlying arithmetic of the number two introduces a unique complexity that requires an extra component to achieve the same level of flexibility.

For grids that are even larger, with four or more rows and columns, the researchers established clear upper limits on the number of terms needed. They demonstrated that for systems based on odd prime numbers, three terms are always sufficient to build any matrix, regardless of how large the grid becomes. This result is significant because it suggests that the complexity of the grid does not require an increasing number of building blocks once a certain size is reached. For systems based on the number two, the limit is four terms. The researchers arrived at these conclusions by developing a step-by-step method to break down any complex matrix into simpler parts, showing how the versatile units can be used to fill in the gaps. Their work provides a comprehensive set of bounds for the requirements of universality in these matrix systems, confirming the maximum number of components needed to ensure that no structure is out of reach.

The implications of this work are rooted in the clarity it brings to the structure of these number systems. By proving that a small, fixed number of terms is sufficient to generate all possible matrices, the researchers have clarified the conditions required for universality in these settings. They have shown that the ability to build any matrix depends entirely on having a sufficient number of versatile coefficients, and that this number is surprisingly small. Whether dealing with two-by-two grids or much larger arrays, the rules are consistent and predictable. This understanding allows mathematicians to treat these complex grids with the same confidence they have when working with simpler numbers, knowing that the fundamental building blocks are sufficient to create any desired outcome. The paper stands as a rigorous proof that in the world of p-adic integers, the power to construct any matrix lies in the hands of just a few well-chosen numbers.

Technical Summary: Integral Quadratic Forms over a Ring of p-adic Integers

Problem Statement
This paper investigates the universality of diagonal integral quadratic forms of the type $\sum_{i=1}^m a_i X_i^2$ over the ring of $n \times n$ matrices with entries in the ring of $p$ -adic integers, denoted as $M_n(\mathbb{Z}_p)$ . A quadratic form is defined as universal over a ring if it can represent every element of that ring. The authors build upon previous work by Lee [1] and Koo and Lee [2], who established necessary and sufficient conditions for universality over $M_2(\mathbb{Z})$ and provided bounds for the function $f(n)$ , defined as the smallest integer $m$ such that a form with $m$ coefficients is universal.

The specific objectives of this paper are:

To establish necessary and sufficient conditions for a diagonal quadratic form to be universal over $M_2(\mathbb{Z}_p)$ for both $p=2$ and odd primes $p$ .
To express matrices over $\mathbb{Z}_p$ as sums of squares based on these conditions.
To define and determine bounds for $f(n)$ for $n \ge 3$ , specifically for $p$ -adic integers $a_1, \dots, a_m$ where at least three coefficients are units.

Methodology
The authors employ algebraic number theory and matrix algebra over local rings. The methodology proceeds in three stages:

Preliminaries and Structure of $\mathbb{Z}_p$ : The paper establishes the foundational properties of the ring of $p$ -adic integers $\mathbb{Z}_p$ as the projective limit of $\mathbb{Z}/p^n\mathbb{Z}$ . It utilizes the $p$ -adic valuation $v_p(x)$ and the structure of units in $\mathbb{Z}_p$ to analyze divisibility and invertibility.
Case $n=2$ :
- For $p=2$ , the authors prove that a form $\sum a_i X_i^2$ is universal over $M_2(\mathbb{Z}_2)$ if and only if at least two of the coefficients $a_i$ are units. The proof involves constructing specific matrix representations and utilizing translational replacements to satisfy systems of equations derived from equating matrix entries.
- For odd primes $p$ , a similar condition is proven: universality holds if and only if at least two coefficients are units. The proofs rely on reducing the problem to the residue field and using the invertibility of units to solve linear systems for the matrix entries.
Case $n \ge 3$ :
- The authors define $f(n)$ for $n \ge 3$ as the smallest $m$ such that any form with at least three unit coefficients is universal over $M_n(\mathbb{Z}_p)$ . This definition sets the premise that the coefficients $a_1, \dots, a_m$ must include at least three units; the goal is then to find the minimal $m$ satisfying this condition.
- To bound $f(n)$ , the paper adapts techniques from Koo and Lee [2]. It introduces specific matrix structures, including the matrix unit $J$ (a shift matrix) and its powers.
- Lemmas 3.1–3.3 establish that any matrix $S \in M_n(\mathbb{Z}_p)$ can be reduced, via subtraction of a square term $a_1 X_1^2$ , to a form similar to $J^2 + \begin{bmatrix} 0 & P \\ 0 & Q \end{bmatrix}$ .
- Lemmas 3.4 and 3.5 address the decomposition of the remaining block matrix. For odd primes, Lemma 3.4 shows that the residual matrix can be represented as a sum of two squares ( $a_2 X_2^2 + a_3 X_3^2$ ) provided $a_2, a_3$ are units. For $p=2$ , Lemma 3.5 demonstrates that three squares ( $a_2 X_2^2 + a_3 X_3^2 + a_4 X_4^2$ ) are required, necessitating an additional unit coefficient.

Key Contributions and Results
The paper provides the following specific results:

Universality over $M_2(\mathbb{Z}_p)$ :
- Theorem 2.1 & 2.3: A diagonal quadratic form $\sum_{i=1}^m a_i X_i^2$ is universal over $M_2(\mathbb{Z}_p)$ if and only if at least two of the coefficients $a_i$ are units in $\mathbb{Z}_p$ . This holds for both $p=2$ and odd primes.
- Corollaries 2.2 & 2.4: Consequently, any matrix in $M_2(\mathbb{Z}_p)$ is either a square or a sum of two squares.
Bounds on $f(n)$ for $n \ge 3$ :
- Theorem 3.1: For $n=3$ , $f(3) \le 4$ .
- Theorem 3.2: For $n \ge 4$ $n \geq 4$ :
  - If $p$ is an odd prime, $f(n) \le 3$ .
  - If $p = 2$ , $f(n) \le 4$ .
- Corollary 3.2: These bounds imply that any matrix in $M_n(\mathbb{Z}_p)$ for $n \ge 4$ can be expressed as a sum of at most three squares (for odd $p$ ) or four squares (for $p=2$ ), provided the coefficients of the quadratic form include the requisite number of units (specifically, at least three units as per the definition of $f(n)$ ).

Significance and Claims
The paper claims to extend the known results on the universality of quadratic forms from the ring of integers $\mathbb{Z}$ to the ring of $p$ -adic integers $\mathbb{Z}_p$ . By establishing precise necessary and sufficient conditions for $n=2$ and deriving upper bounds for $n \ge 3$ , the work refines the understanding of how many squares are required to represent arbitrary matrices in these local rings. The authors explicitly state that their proofs for $n \ge 4$ follow the methodology of Koo and Lee [2] but adapt them to the $p$ -adic context, specifically handling the distinct algebraic properties of $p=2$ versus odd primes. The paper does not propose new applications or experimental frameworks but focuses on the theoretical characterization of these quadratic forms.

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