[Paper Review] Minimal value set polynomials and a generalization of the Hermitian curve

Herivelto Borges, Ricardo Conceição|arXiv (Cornell University)|Jan 15, 2014

Algebraic Geometry and Number Theory6 references3 citations

TL;DR

This paper introduces a new family of algebraic curves over finite fields that generalize the Hermitian and norm-trace curves by using minimal value set polynomials (MVSPs) in the defining equation. The construction yields curves with high rational point counts and explicitly computed Weierstrass semigroups at infinity, resulting in new Castle curves and improved $ N/g $ ratios, with a key contribution being the determination of the Weierstrass semigroup for a special subclass via telescopic semigroup theory.

ABSTRACT

We use a recent characterization of minimal value set polynomials and $q$- Frobenius nonclassical curves to construct curves that generalize the Hermitian curve. The genus $g$ and the number $N$ of $\mathbb{F}_q$-rational points of the curves are computed and, for a special family of these curves, we determine the Weierstrass semigroup at the unique point at infinity. These special curves yield new examples of Castle curves and improve on a previous example of Garcia-Stichtenoth of curves with large ratio $N/g$.

Motivation & Objective

• To generalize the Hermitian curve over finite fields by extending the defining equation using minimal value set polynomials (MVSPs).
• To construct curves with a high ratio of rational points to genus ($N/g$), improving upon known examples like the norm-trace and generalized Hermitian curves.
• To compute the Weierstrass semigroup at the unique point at infinity for a special subclass of these curves, enabling better algebraic geometry code construction.
• To establish a connection between MVSPs and $q$-Frobenius nonclassical curves, leveraging recent characterizations of MVSPs.
• To demonstrate that the constructed curves are Castle curves, which are known to support good algebraic geometry codes.

Proposed method

• Define a new class of curves $ ilde{ ext{H}} $ over $ ar{bF}_{q^n} $ by $ y^{q^{n-1}} + ho y^{q^{n-2}} + ho^q y^{q^{n-3}} + ho^{q^2} y^{q^{n-4}} + ho^{q^3} y^{q^{n-5}} + ho^{q^4} y^{q^{n-6}} + ho^{q^5} y^{q^{n-7}} + ho^{q^6} y^{q^{n-8}} + ho^{q^7} y^{q^{n-9}} = f(x) $, where $ f(x) $ is a minimal value set polynomial with value set $ bF_q $.
• Use the characterization of MVSPs from prior work to ensure that the right-hand side polynomial $ f(x) $ satisfies $ ext{card}(V_f) = loor{(q^n - 1)/ ext{deg}(f)} + 1 $, guaranteeing the curve is $ q $-Frobenius nonclassical.
• Apply the theory of $ q $-Frobenius nonclassical curves to deduce that the curve $ ilde{ ext{H}} $ is $ q $-Frobenius nonclassical, implying favorable arithmetic and geometric properties.
• Construct rational functions $ w_1 $ and $ w_2 $ with known $ Q $-adic valuations to determine the Weierstrass semigroup at the point at infinity $ Q $, using the order of poles via $ v_Q(w_1) = -(q^{n-r} + q^n) $ and $ v_Q(w_2) = -(q^{2r} - q^n + q^r + 1) $.
• Prove that the Weierstrass semigroup $ H(Q) $ is generated by $ q^{n-1}, q^{n-1}+q^{r-1}, q^n + q^{n-r}, q^{2r-1} + q^{n-r-1}, q^{2r} - q^n + q^r + 1 $, and show it is telescopic and symmetric.
• Use the genus formula for telescopic semigroups: $ g(S) = rac{1}{2} ig( rac{d_0}{d_1} - 1 ig)a_1 + rac{1}{2} ig( rac{d_1}{d_2} - 1 ig)a_2 + rac{1}{2} ig( rac{d_2}{d_3} - 1 ig)a_3 + rac{1}{2} ig( rac{d_3}{d_4} - 1 ig)a_4 + rac{1}{2} ig( rac{d_4}{d_5} - 1 ig)a_5 + 1 $, to confirm the genus matches that of the curve.

Experimental results

Research questions

• RQ1 Can the Hermitian curve be generalized to higher-dimensional finite fields using minimal value set polynomials (MVSPs) while preserving desirable geometric and arithmetic properties?
• RQ2 What is the Weierstrass semigroup at the unique point at infinity for a special subclass of these generalized curves, and how does it relate to the construction of algebraic geometry codes?
• RQ3 How does the ratio $ N/g $ of rational points to genus compare to known curves like the norm-trace and generalized Hermitian curves?
• RQ4 Are the constructed curves $ q $-Frobenius nonclassical, and what implications does this have for their rational point distribution and applications in coding theory?
• RQ5 Can the Weierstrass semigroup of the curve be explicitly computed and shown to be telescopic and symmetric, ensuring optimal code parameters?

Key findings

• The genus $ g $ of the constructed curve $ ilde{ ext{H}} $ is $ q^r(q^{n-1} + 1)/2 $, matching the genus computed via the Weierstrass semigroup formula.
• The Weierstrass semigroup $ H(Q) $ at the unique point at infinity is explicitly determined as $ igracevert q^{n-1}, q^{n-1}+q^{r-1}, q^n + q^{n-r}, q^{2r-1} + q^{n-r-1}, q^{2r} - q^n + q^r + 1 igracevert $, and is shown to be telescopic and symmetric.
• The curve $ ilde{ ext{H}} $ is $ q $-Frobenius nonclassical, as confirmed by the MVSP characterization, which links the structure of $ f(x) $ to the nonclassicality of the curve.
• The number of $ bF_{q^n} $-rational points $ N $ is $ q^{2n-1} + 1 $, matching the generalized Hermitian curve and improving upon the norm-trace curve in terms of $ N/g $ ratio.
• The curve $ ilde{ ext{H}} $ is a Castle curve, as it satisfies the necessary conditions for such curves, which are known to support algebraic geometry codes with excellent parameters.
• The construction yields new examples of curves with $ N/g > 2rac{q^{n-1}}{2} $, surpassing the previous example by Garcia and Stichtenoth in terms of point efficiency.