Positivity preservers over finite fields II

Abstract: We say that a matrix over a finite field $\mathbb{F}_q$ is positive definite if it is symmetric and each of its leading principal minors is a nonzero square in $\mathbb{F}_q$. In previous work of the authors [J. Algebra, 2025], the entrywise positivity preservers on $M_n(\mathbb{F}_q)$ were classified for every $n\geq 2$, with one remaining case: $n=2$, $q\equiv 1\pmod 4$, and $q$ not a square. We settle this case by proving that every positivity preserver on $M_2(\mathbb{F}_q)$ is injective on the set $\mathbb{F}_q<sup>+$ of nonzero squares whenever $q\equiv 1\pmod 4$. The proof combines an idempotent reduction of positivity preservers with a well-known property of quadratic characters. This yields the complete classification of entrywise positivity preservers over every finite field and in every fixed dimension.

• The paper proves that every positivity preserver on 2×2 matrices over F_q with q ≡ 1 mod 4 is automatically injective on the nonzero squares, eliminating the final unresolved case.
• The proof uses quadratic-character separation and an idempotent iterate of the finite-field map to derive a contradiction from any failure of injectivity, without relying on Paley graph clique classifications.
• The complete classification shows that for odd q and n ≥ 2, preservers are exactly maps f(x)=c x^{p^j} with c a nonzero square, while dimension two in even characteristic also permits bijective monomials.

Background and problem

The study of entrywise transforms preserving positive definiteness is a classical topic in matrix analysis, originating with Pólya and Szegő in 1925. This paper addresses the finite-field analogue, where a matrix $A \in M_n(F_q)$ is declared positive definite if it is symmetric and all its leading principal minors are nonzero squares in $F_q$. A map $f: F_q \to F_q$ is an entrywise positivity preserver on $M_n(F_q)$ if $f[A]$ is positive definite whenever $A$ is.

Prior work by the same authors (the "GGVY" paper in J. Algebra, 2025) classified all preservers for $n \geq 3$ — they are exactly the positive multiples of field automorphisms $x \mapsto c\,x^{p^j}$ with $c \in F_q^+$ — and settled the two-dimensional case when $q$ is even, when $F_q$0, and when $F_q$1 is an odd square. The sole remaining case was:

$F_q$2

For this case, GGVY isolated a reduction strategy (their Proposition 5.8): if $F_q$3 preserves positivity on $F_q$4, satisfies $F_q$5, and is injective on $F_q$6, then $F_q$7 on all of $F_q$8. The injectivity hypothesis was the obstruction. Its verification via the GGVY machinery required structural results on maximal cliques of Paley graphs of square order; for nonsquare $F_q$9, such results are unavailable, and even the asymptotic clique number of Paley graphs of nonsquare order is an open problem. The present note removes this barrier entirely.

Main result: automatic injectivity

Theorem. Let $f: F_q \to F_q$0. If $f: F_q \to F_q$1 preserves positive definiteness on $f: F_q \to F_q$2, then $f: F_q \to F_q$3 is injective.

Two ingredients drive the proof. First, any preserver maps $f: F_q \to F_q$4 into itself, since $f: F_q \to F_q$5 is positive definite for $f: F_q \to F_q$6 and the first leading principal minor of $f: F_q \to F_q$7 must be a square. Second, a separation property of quadratic characters (proved in GGVY via strong regularity of Paley graphs): distinct $f: F_q \to F_q$8 admit some $f: F_q \to F_q$9 with $M_n(F_q)$0 and $M_n(F_q)$1.

The proof then proceeds by contradiction and is notable for its economy. If $M_n(F_q)$2 for distinct $M_n(F_q)$3, finiteness of $M_n(F_q)$4 guarantees that some iterate $M_n(F_q)$5 is idempotent ($M_n(F_q)$6), and idempotent maps preserve positivity whenever $M_n(F_q)$7 does. Setting $M_n(F_q)$8 and applying $M_n(F_q)$9 to the two positive definite matrices $f[A]$0 and $f[A]$1 forces both $f[A]$2 and $f[A]$3 into $f[A]$4, where $f[A]$5. Idempotence gives $f[A]$6, so $f[A]$7. A short case analysis on the quadratic character of $f[A]$8 produces a positive definite matrix $f[A]$9 whose image under $A$0 is the singular matrix $A$1 — a contradiction.

This argument works uniformly for every $A$2 and does not invoke clique structure at all. In particular, it bypasses the subfield analysis that made the square-order case in GGVY comparatively intricate, and it sidesteps the open clique-number problem for Paley graphs of nonsquare order. The device of passing to an idempotent iterate is the key technical innovation: it converts eventual periodicity of self-maps of a finite set into a rigidity constraint compatible with the determinant sign conditions.

Complete classification

Combining the injectivity theorem with Proposition 5.8 of GGVY yields the full classification over every finite field and every fixed dimension. For $A$3 and $A$4, $A$5 preserves positive definiteness on $A$6 precisely in the following cases:

Case	Preservers
$A$7	$A$8
$A$9, $n \geq 3$0 even	bijective monomials $n \geq 3$1, $n \geq 3$2
$n \geq 3$3, $n \geq 3$4 even	$n \geq 3$5, $n \geq 3$6
$n \geq 3$7, $n \geq 3$8 odd	$n \geq 3$9, $x \mapsto c\,x^{p^j}$0

The proof for $x \mapsto c\,x^{p^j}$1, $x \mapsto c\,x^{p^j}$2 normalizes $x \mapsto c\,x^{p^j}$3 by $x \mapsto c\,x^{p^j}$4 with $x \mapsto c\,x^{p^j}$5, applies the injectivity theorem to $x \mapsto c\,x^{p^j}$6, and invokes Proposition 5.8 to obtain $x \mapsto c\,x^{p^j}$7. Conversely, each listed form plainly preserves positivity in all dimensions. A structural consequence worth noting: in odd characteristic, the dimension-two classification coincides exactly with the classification for all $x \mapsto c\,x^{p^j}$8, whereas in even characteristic dimension two admits strictly more preservers (all permutation monomials rather than only Frobenius powers).

Limitations and open questions

The result is complete for fixed-dimension entrywise preservers over finite fields, but several boundaries remain. First, the classification is pointwise in $x \mapsto c\,x^{p^j}$9; whether uniform characterizations exist across varying dimensions or field towers is not addressed. Second, the semidefinite analogue — preservers of positive semidefinite matrices over finite fields, studied recently by Ayyer and Prasad — is not treated here, and the leading-minor definition used does not directly transfer. Third, the paper's method is intrinsically finite: the idempotent-iterate trick relies on finiteness of $c \in F_q^+$0 and offers no evident extension to infinite fields of positive characteristic, where the corresponding classification problem remains open. Finally, while the proof avoids Paley graph clique structure, the deeper question of clique numbers of Paley graphs of nonsquare order — which motivated the difficulty in the first place — is untouched and remains open.

Conclusion

This paper closes the last outstanding case in the classification of entrywise positivity preservers over finite fields. Its central contribution is a short, uniform proof that any $c \in F_q^+$1 preserver with $c \in F_q^+$2 is automatically injective on the nonzero squares, obtained by combining iterates of the preserver (yielding an idempotent reduction) with a quadratic-character separation lemma. The resulting classification shows that, in odd characteristic, positivity preservers are exactly positive scalar multiples of Frobenius maps in every dimension $c \in F_q^+$3, unifying the two-dimensional theory with the previously known higher-dimensional case.