Minimal value set binomials and Frobenius nonclassical curves

Technical Summary: Minimal Value Set Binomials and Frobenius Nonclassical Curves

Problem Statement
This paper addresses two interconnected problems in finite field theory and algebraic geometry. First, it seeks to characterize minimal value set polynomials (MVSPs) specifically within the class of binomials (polynomials with two terms) over a finite field $\mathbb{F}_q$. An MVSP is defined as a polynomial $F \in \mathbb{F}_q[x]$ that attains the theoretical lower bound for the cardinality of its value set, given by $\lceil (q-1)/\deg F \rceil + 1$. Second, utilizing the classification of these binomials, the paper aims to classify irreducible quadrinomial curves (curves defined by polynomials with four terms) that are $\mathbb{F}_q$-Frobenius nonclassical. A curve is Frobenius nonclassical if the image of any smooth point under the $\mathbb{F}_q$-Frobenius morphism lies on its tangent line. The authors focus on curves with separated variables, defined by equations of the form $f(x) = g(y)$.

Methodology
The authors employ a combination of algebraic number theory and the theory of algebraic curves:

1. Characterization of Binomials: The authors analyze binomials of the form $F(x) = x^a + \beta x^b$. They utilize a structural theorem for MVSPs (Theorem 2.2) which relates the value set structure to the existence of specific polynomials and Hasse derivatives. The analysis is divided into three cases based on the size of the value set $|V_F|$:

   • $|V_F| = 1$: Direct application of the definition of MVSPs and divisibility properties.
   • $|V_F| = 2$: Analysis using recurrence relations of the polynomial evaluated at powers of a primitive element.
   • $|V_F| > 2$: Application of Theorem 2.2 to derive constraints on the exponents and coefficients, specifically analyzing the cancellation of terms in high-degree expansions and the properties of the norm map in field extensions.

2. Classification of Curves: The authors leverage a known connection (Proposition 2.5, from [3]) stating that the irreducible components of a curve $f(x) = g(y)$ are Frobenius nonclassical if and only if $f$ and $g$ are MVSPs with the same value set (under certain conditions).

   • They categorize quadrinomial curves with separated variables into three types based on their affine equation forms: (i) $k_1 x^{m_1} + k_2 x^{m_2} + k_3 y^{n_1} + y^{n_2} = 0$, (ii) $k_1 x^{m_1} + k_2 x^{m_2} + k_3 y^{n_1} + 1 = 0$, and (iii) $k_1 x^{m_1} + k_2 x^{m_2} + k_3 x^{m_3} + y^{n_4} = 0$.
   • For each type, they substitute the classified binomials (and derived trinomials for type (iii)) into the curve equation.
   • They apply criteria for irreducibility (Proposition 5.3) and Frobenius nonclassicality (Proposition 5.1) to filter the results, distinguishing between reducible and irreducible components.

Key Contributions and Results

1. Classification of Minimal Value Set Binomials (Theorem A)
The paper provides a complete classification of binomials over $\mathbb{F}_q$ that are MVSPs. Up to multiplication by a nonzero element, a binomial is an MVSP if and only if it takes one of the following forms:

• $x^a + \beta$ where $\beta \in \mathbb{F}_q^*$ and $a \mid (q-1)$.
• $x^{b+\ell(q-1)} - x^b$ for some $b, \ell \in \mathbb{N}$ (Value set size 1).
• Specific forms involving $q=2^{2n}$ or $p>2$ with exponents related to $(q-1)/3$ or $(q-1)/2$ (Value set size 2).
• $x^{p^\ell t} + \beta x^t$ and $x^{2t} + \beta x^t$ with specific constraints on $t$, $\beta$, and the norm of $\beta$ (Value set size $>2$).

These results are noted to be consistent with Conjecture 1.1 from [6].

2. Classification of Frobenius Nonclassical Quadrinomial Curves (Theorem B)
Using the binomial classification and the connection to separated variable curves, the authors classify irreducible quadrinomial curves $C$ over $\mathbb{F}_q$ that are Frobenius nonclassical. The curves are given by specific plane equations, including:

• Type (i): Equations of the form $y^{d+q-1} - y^d = \alpha x^{b+q-1} - \alpha x^b$.
• Type (ii): Equations involving $y^{q-1}$ and specific trinomial combinations of $x$ (e.g., $y^{q-1} = x^{2(q-1)/3} + x^{(q-1)/3} + 1$ for $F_4 \subseteq \mathbb{F}_q$).
• Type (iii): More complex equations involving powers of $p$, such as $y^{(p^{2\ell}+p^\ell+1)t} = x^{p^{2\ell}t} + x^{p^\ell t} + x^t$ and variations with coefficients in subfields.

The authors explicitly note that many curves derived in the process are reducible (often products of Fermat curves and lines) and are excluded from the final classification of irreducible curves in Theorem B.

Significance and Claims
The paper claims to be the first work to classify MVSPs by fixing the number of monomials (specifically binomials). The authors position this as a new approach to the broader problem of classifying MVSPs, which has previously been studied by fixing the field size or constructing MVSPs from known polynomials.

Furthermore, the paper extends the classification of Frobenius nonclassical curves from trinomial curves (previously classified by the second author in [10]) to quadrinomial curves. The results provide a definitive list of irreducible quadrinomial curves with separated variables that possess the Frobenius nonclassical property, a class of curves known for potentially having a high number of rational points (exceeding the Hasse-Weil bound for classical curves). The findings align with and provide evidence for Conjecture 1.1 and Theorem 2.8 in [6].