Positivity preservers over finite fields II

Technical Summary: Positivity Preservers Over Finite Fields II

Problem Statement
This paper addresses the classification of entrywise positivity preservers on the space of $n \times n$ matrices over a finite field $\mathbb{F}_q$, denoted $M_n(\mathbb{F}_q)$. A matrix $A \in M_n(\mathbb{F}_q)$ is defined as positive definite if it is symmetric and all its leading principal minors are nonzero squares in $\mathbb{F}_q$ (elements of $\mathbb{F}_q^+$). A function $f: \mathbb{F}_q \to \mathbb{F}_q$ is an entrywise positivity preserver if the matrix $f[A] = (f(a_{ij}))$ is positive definite whenever $A$ is.

While previous work by the authors (published in J. Algebra, 2025) had successfully classified these preservers for all dimensions $n \ge 3$ and for the case $n=2$ when $q$ is even, $q \equiv 3 \pmod 4$, or $q$ is an odd square, one specific case remained unresolved:

• The Open Case: $n = 2$, $q \equiv 1 \pmod 4$, and $q$ is not a perfect square.

In the prior work, a strategy was isolated (Proposition 5.8 in [5]) suggesting that if $f$ preserves positivity on $M_2(\mathbb{F}_q)$ with $q \equiv 1 \pmod 4$, and if $f$ is normalized such that $f(1)=1$, then $f$ must be a power map ($f(x) = x^{p^j}$) provided that the restriction of $f$ to the set of nonzero squares $\mathbb{F}_q^+$ is injective. The missing link was proving that this injectivity is automatic for all such $q$.

Methodology
The authors resolve the open case by proving that any positivity preserver on $M_2(\mathbb{F}_q)$ (where $q \equiv 1 \pmod 4$) is necessarily injective on $\mathbb{F}_q^+$. The proof employs a novel approach that avoids the structural complexities of Paley graphs that hindered previous attempts for non-square $q$.

1. Idempotent Reduction: The authors consider the sequence of iterates of the function $f$ (i.e., $f, f^{(2)}, f^{(3)}, \dots$). Since $\mathbb{F}_q$ is finite, this sequence eventually becomes periodic. They select an iterate $h = f^{(m)}$ such that $h$ is an idempotent positivity preserver (i.e., $h \circ h = h$).
2. Contradiction via Quadratic Characters: Assuming $h$ is not injective on $\mathbb{F}_q^+$, there exist distinct $a, b \in \mathbb{F}_q^+$ such that $h(a) = h(b) = s$.
   • They utilize a known property of quadratic characters (Lemma 2.1): for distinct $a, b \in \mathbb{F}_q^+$, there exists a non-square $z \in \mathbb{F}_q^-$ such that $a-z \in \mathbb{F}_q^+$ and $b-z \in \mathbb{F}_q^-$.
   • They construct two specific positive definite matrices, $A_1$ and $A_2$, involving $a, b, z$.
   • By applying the idempotent map $h$ to these matrices, they derive constraints on the value $v = h(z)$. Specifically, the preservation of positive definiteness forces $v$ to be a square ($v \in \mathbb{F}_q^+$).
   • They then construct a third matrix $C$ using $v$ and $z$. Depending on whether $v-z$ is a square or non-square, $C$ is shown to be positive definite.
   • However, applying $h$ to $C$ yields a singular matrix (a matrix with a zero determinant), which contradicts the definition of a positivity preserver.
3. Uniformity: This proof works uniformly for all $q \equiv 1 \pmod 4$, bypassing the need for the "clique structure" arguments used in the square case, which rely on subfield structures not available for non-square orders.

Key Contributions and Results
The primary contribution is the proof of Theorem 1.1, which establishes that for $q \equiv 1 \pmod 4$, any function $f$ preserving positive definiteness on $M_2(\mathbb{F}_q)$ restricts to an injective map on $\mathbb{F}_q^+$.

This result, combined with the previous classification in [5], leads to the Complete Classification (Theorem 1.2) of entrywise positivity preservers for every finite field and every fixed dimension $n$:

• Case $n=1$: $f(\mathbb{F}_q^+) \subseteq \mathbb{F}_q^+$.
• Case $n=2$, $q$ even: $f$ is a bijective monomial $f(x) = cx^m$ where $\gcd(m, q-1)=1$.
• Case $n \ge 3$, $q$ even: $f(x) = cx^{2^j}$.
• Case $n \ge 2$, $q$ odd: $f(x) = cx^{p^j}$ for some $c \in \mathbb{F}_q^+$ and $0 \le j \le k-1$ (where $q=p^k$).

Notably, the authors observe that in odd characteristic, the classification for dimension $n=2$ is identical to the classification for all dimensions $n \ge 3$.

Significance and Claims
The paper claims to settle the final remaining case in the classification of entrywise positivity preservers over finite fields. By proving the automatic injectivity of these maps on the set of squares, the authors unify the theory across all dimensions and field characteristics.

The significance of the work lies in:

1. Completing the Classification: It provides a definitive answer to a problem that had been partially solved, leaving only the specific non-square, $n=2$ case open.
2. Methodological Innovation: The proof introduces an "idempotent reduction" technique. This approach successfully circumvents the difficulty of analyzing the clique numbers of Paley graphs of non-square order, a notoriously open problem in combinatorics.
3. Unification: It demonstrates that the structural constraints on positivity preservers in dimension $n=2$ (for odd $q$) are no more complex than in higher dimensions, contrary to what might be expected given the specific challenges of the $n=2$ case.

The authors explicitly state that their proof does not rely on the special subfield structure available when $q$ is a square, making the result robust for all $q \equiv 1 \pmod 4$.