On planar functions over $\mathbb{F}_{q^3}$

Abstract: Let $\mathbb{F}<em>q$ denote the finite field of order $q$. For $q$ odd, we investigate the planarity over $\mathbb{F}</em>{q<sup>3}$ of the family $$f_{E,A,B,C,D}(X) := EX²⁺ AX^{q+1}+ BX^{{q^{2+1}+CX^{2q}}} +DX^{{2q^2}\in} \mathbb{F}_{q}[X].$$ Using results from the theory of q-polynomials, we establish conditions under which these polynomials are planar functions. In particular, we provide characterizations for the planarity property and present new families of planar trinomials, quadrinomials, and pentanomials.

• The paper characterizes planarity for a five-term polynomial family by expressing its derivatives as linearized polynomials and requiring the associated 383 Dickson matrix to be nonsingular for every nonzero field element.
• The paper develops a factorization framework using root-free linearized trinomials, producing new planar trinomials, quadrinomials, and pentanomials over F_{q^3}, including families valid for all odd prime powers q.
• The paper gives an if-and-only-if criterion for a two-parameter pentanomial family, while showing that its quadrinomial construction requires an element BF_q satisfying Bb3=4.

Overview

The paper studies planar functions over $\mathbb{F}_{q^3}$ for odd prime powers $q$, focusing on the five-term family

$$f_{E,A,B,C,D}(X) = EX^2 + AX^{q+1} + BX^{q^2+1} + CX^{2q} + DX^{2q^2} \in \mathbb{F}_q[X].$$

Recall that $f$ is planar (perfect nonlinear) if for every $\epsilon \neq 0$ the derivative $f(X+\epsilon)-f(X)$ is a permutation of $\mathbb{F}_{q^3}$. The authors' central contribution is a complete coefficient-level characterization of planarity for this family, together with a general factorization-based construction technique from which they derive new planar trinomials, quadrinomials, and pentanomials. The work situates itself alongside recent classification efforts such as Chen–Mesnager's treatment of trace-plus-quadratic planar functions over cubic extensions and Chan–Xiong's classification of planar quadrinomials over $\mathbb{F}_{q^2}$.

Method: linearized derivatives and Dickson matrices

The key structural observation is that for this family the derivative polynomial

$$F_\epsilon(X) := f_{E,A,B,C,D}(X+\epsilon) - f_{E,A,B,C,D}(X) - f_{E,A,B,C,D}(\epsilon)$$

is always a $q$-polynomial (linearized polynomial):

$q$0

Planarity therefore reduces to nonsingularity of the associated $q$1 Dickson ($q$2-circulant) matrix for all $q$3. This yields Proposition 3.1, an explicit but unwieldy necessary-and-sufficient condition: a certain quartic expression in $q$4 with coefficients built from $q$5 must be nonzero on all of $q$6. While exact, this condition is not directly verifiable in general, which motivates the constructive machinery developed next.

A recurring tool is Lemma 2.1-style criterion: the trinomial $q$7 permutes $q$8 (equivalently, has no nonzero root there) if and only if $q$9. This circulant-determinant identity is used repeatedly to certify root-freeness of candidate factors.

A product-factorization framework

Two technical propositions establish that two specific structured matrices $$f_{E,A,B,C,D}(X) = EX^2 + AX^{q+1} + BX^{q^2+1} + CX^{2q} + DX^{2q^2} \in \mathbb{F}_q[X].$$0 and $$f_{E,A,B,C,D}(X) = EX^2 + AX^{q+1} + BX^{q^2+1} + CX^{2q} + DX^{2q^2} \in \mathbb{F}_q[X].$$1 have determinants that never vanish on $$f_{E,A,B,C,D}(X) = EX^2 + AX^{q+1} + BX^{q^2+1} + CX^{2q} + DX^{2q^2} \in \mathbb{F}_q[X].$$2: in the first case the determinant factors as $$f_{E,A,B,C,D}(X) = EX^2 + AX^{q+1} + BX^{q^2+1} + CX^{2q} + DX^{2q^2} \in \mathbb{F}_q[X].$$3, each factor being root-free by the trinomial criterion; in the second it equals $$f_{E,A,B,C,D}(X) = EX^2 + AX^{q+1} + BX^{q^2+1} + CX^{2q} + DX^{2q^2} \in \mathbb{F}_q[X].$$4 where $$f_{E,A,B,C,D}(X) = EX^2 + AX^{q+1} + BX^{q^2+1} + CX^{2q} + DX^{2q^2} \in \mathbb{F}_q[X].$$5 factors into three scalar multiples of cyclic shifts of $$f_{E,A,B,C,D}(X) = EX^2 + AX^{q+1} + BX^{q^2+1} + CX^{2q} + DX^{2q^2} \in \mathbb{F}_q[X].$$6, again root-free.

These computations motivate Theorem 4.1, the paper's main construction principle. If one can find three root-free trinomials $$f_{E,A,B,C,D}(X) = EX^2 + AX^{q+1} + BX^{q^2+1} + CX^{2q} + DX^{2q^2} \in \mathbb{F}_q[X].$$7 whose coefficients satisfy three symmetric compatibility conditions (equalities among products of the $$f_{E,A,B,C,D}(X) = EX^2 + AX^{q+1} + BX^{q^2+1} + CX^{2q} + DX^{2q^2} \in \mathbb{F}_q[X].$$8), then any solution $$f_{E,A,B,C,D}(X) = EX^2 + AX^{q+1} + BX^{q^2+1} + CX^{2q} + DX^{2q^2} \in \mathbb{F}_q[X].$$9 of an explicit four-equation linear-ish system forces the derivative determinant to factor exactly as $f$0, hence nonzero — so $f$1 is planar. A corollary specializes this to the case $f$2, $f$3, reducing the input data to a single root-free trinomial.

This framework converts planarity constructions into solving finite systems of equations over $f$4, a substantially more tractable problem than direct verification of the determinant condition.

Explicit families

Applying the framework with particular choices of root-free factors yields the following results:

Polynomial	Condition	Type
$f$5	$f$6	trinomial
$f$7	none (requires $f$8 solvable in $f$9)	quadrinomial
$\epsilon \neq 0$0	$\epsilon \neq 0$1	pentanomial
$\epsilon \neq 0$2	none	pentanomial
$\epsilon \neq 0$3	none	pentanomial

Several remarks are in order. The trinomial family is genuinely two-parameter (any $\epsilon \neq 0$4 on the affine cubic surface $\epsilon \neq 0$5), and its proof is the degenerate case of the framework obtained by taking the three monomial factors $\epsilon \neq 0$6. The quadrinomial result carries a caveat the authors state explicitly: it requires an element $\epsilon \neq 0$7 with $\epsilon \neq 0$8, so it applies only when $\epsilon \neq 0$9 or when $f(X+\epsilon)-f(X)$0 is a cube in $f(X+\epsilon)-f(X)$1; the statement as given is conditional on this solvability. The two-parameter pentanomial family is the strongest result in the sense of being an if-and-only-if characterization: planarity holds precisely when $f(X+\epsilon)-f(X)$2. The final two pentanomials are proved directly via the determinant computations of the two matrix propositions rather than through the general system, since their coefficient vectors match those special structured cases.

All families are stated uniformly for arbitrary odd prime powers $f(X+\epsilon)-f(X)$3, with coefficients in the base field $f(X+\epsilon)-f(X)$4 only — a restriction that keeps the Dickson matrices circulant-symmetric and is essential to the factorization arguments.

Limitations and open questions

The paper's framework is sufficient rather than exhaustive in most instances: Theorem 4.1 gives a sufficient condition for planarity, and only the two-parameter pentanomial family receives a full characterization. Whether the determinant condition of Proposition 3.1 admits solutions outside the factorization-compatible regime is not addressed. The quadrinomial family depends on the cube $f(X+\epsilon)-f(X)$5 existing in $f(X+\epsilon)-f(X)$6, leaving the complementary case untreated. Additionally, the paper does not investigate equivalence classes of the new functions under the standard equivalences for planar maps (affine equivalence, isotopy), so it remains open whether these families are new up to equivalence or linearly equivalent to known Dembowski–Ostrom polynomials; nor are the associated commutative semifields or projective planes examined.

Conclusion

The paper provides a clean reduction of the planarity problem for a natural five-term family over $f(X+\epsilon)-f(X)$7 to nonsingularity of an explicit $f(X+\epsilon)-f(X)$8 Dickson matrix, and develops a factorization method — based on root-free linearized trinomials satisfying symmetric coefficient identities — that systematically produces planar polynomials. The concrete outputs include a two-parameter planar trinomial family governed by the Markov-type surface equation $f(X+\epsilon)-f(X)$9, a characterized two-parameter pentanomial family, and several sporadic quadrinomials and pentanomials valid for all odd $\mathbb{F}_{q^3}$0. The approach suggests that extending the same Dickson-matrix analysis to degree-$\mathbb{F}_{q^3}$1 analogues over $\mathbb{F}_{q^3}$2, or classifying the new functions up to equivalence, are natural next problems.