Computing Degree Sets: Methods Overview

• Method for Computing Degree Sets is a framework that explicitly determines the set of occurring degrees in algebraic, geometric, and combinatorial objects.
• It uses a stepwise algorithm involving enumeration of closed points, semigroup construction, and scaling by residue degrees to capture complex arithmetic and geometric data.
• The approach extends to applications in superelliptic curves, symbolic dynamics, and commutative algebra, providing deep insights into structural and computational properties.

A method for computing degree sets refers to a framework or explicit algorithmic procedure for determining the set of occurring degrees of specific geometric, algebraic, or combinatorial objects—such as varieties over local fields, algebraic curves, or structures arising in symbolic dynamics and commutative algebra. Degree sets encode information such as possible residual field extensions for rational points, representable degrees of closed points on algebraic varieties, or the set of degrees realized by vertices in a graph or fibers in a coding map. Such sets often have deep arithmetic, geometric, or combinatorial significance.

1. Degree Sets in Arithmetic Geometry

Degree sets in arithmetic geometry concern the set of degrees of closed points on a variety $W$ over the field of fractions $K$ of a Henselian discrete valuation ring $R$ with perfect residue field $k$. The degree set of $W$ is defined as

$$D(W) := \{\,\deg_K(P) : P \in W \text{ closed point}\,\}\subset\mathbb N$$

The index $\mathrm{ind}(W)$ is the greatest common divisor of $D(W)$.

Recent results show that if $W/K$ is smooth, geometrically irreducible, projective, and admits a regular, proper, flat model $X\to S=\operatorname{Spec}R$ with special fiber $X_s$ a strict normal crossings (SNC) divisor, then the entire degree set $D(W)$ can be computed from purely combinatorial data of $X_s$ as follows: $$D(W) = \bigcup_{x\in X_s(\bar{k})} \deg_k(x) \cdot N(x)$$ where $N(x)$ is the additive semigroup generated by the multiplicities $m_i$ of the irreducible components $E_i$ of $X_s$ containing $x$, and $\deg_k(x) = [k(x):k]$ is the residue field extension degree (Creutz et al., 2023).

This theorem of Creutz–Viray generalizes and refines previous index formulae of Gabber–Liu–Lorenzini, fully capturing the combinatorial nature of degree phenomena for such curves.

2. Algorithmic Procedure for Computing Degree Sets over Henselian Fields

The computation of $D(W)$ under the SNC model hypothesis proceeds as a finite combinatorial algorithm:

1. Enumerate closed points $x$ of $X_s$ (or over finite extensions as needed), noting for each:
   • The set $I(x)=\{i: x\in E_i\}$
   • The multiplicities $\{m_i: i\in I(x)\}$
   • The residue degree $\deg_k(x)$
2. For each $x$, compute the numerical semigroup $N(x)$ generated by $\{m_i\}$. This is the set of nonnegative integer linear combinations of the $m_i$.
3. Scale $N(x)$ by $\deg_k(x)$ to obtain $T_x = \{\, \deg_k(x)\cdot d : d\in N(x)\}$.
4. Form the union: $D(W) = \bigcup_{x\in X_s} T_x$.

Pseudocode for this computation:

D = set() for each closed point x in X_s: M = [m_i for i in I(x)] Sx = semigroup_generated_by(M) for d in Sx: D.add(deg_k(x) * d) return D

This procedure is polynomial in the size of the dual graph of $X_s$, which is $O(g)$ for a genus $g$ curve, and in the number of semigroup generators, which are usually few (Creutz et al., 2023).

3. Structural and Arithmetic Properties

The structure of $D(W)$ over Henselian fields contrasts sharply with the case over finitely generated fields. For curves over finitely generated fields, $D(W)$ always contains all sufficiently large multiples of the index. In contrast, for curves over $p$-adic fields or other Henselian fields with finite or algebraically closed residue field, $D(W)$ can be highly non-cofinite. For example, there exist genus 2 curves over $\mathbb Q_p$ with index 1 for which $D(W)$ excludes all integers coprime to 6 (Creutz et al., 2023).

Key formulas:

• Local semigroup: $N(x) = \langle m_i : i\in I(x)\rangle_{\mathbb N}$
• Index: $$\mathrm{ind}(W) = \gcd\{ m_i [k(E_i): k] : i = 1, \ldots, n\}$$
• At node $x$ lying on $E_1, E_2$ with multiplicities $m_1, m_2$:

$$\deg(P) = \deg_k(x)(a m_1 + b m_2) \quad (a,b\in \mathbb N)$$

This combinatorial description yields explicit control of excluded primes and rich behavior depending on the geometry of the special fiber.

4. Extensions and Special Cases: Superelliptic Covers

For superelliptic curves $C/K$ with equations $y^q = F(x)$ over discretely valued Henselian fields with residue characteristic coprime to $q$ (a prime), an explicit method has been developed for computing degree sets. The computation relies on the clustering of roots of $F(x)$ in the $v$-adic metric, Galois actions, and associated orbit sizes.

The degree set $\mathcal D(C/K)$ is precisely determined by combinatorial congruences involving:

• The number of roots in each cluster ($b_s$)
• A valuation-based sum ($c_s$)
• Galois invariance and orbit sizes of clusters
• Obstructions from values $v(F(0))$, $v(F(\gamma_s))$ at special points

The explicit algorithm assembles $\mathcal D(C/K)$ as a union of arithmetic progressions of the form $$q\mathbb N \cup \bigcup_{s \text{ non-inv}} |\mathcal O(s)|\mathbb N$$, after performing cluster decompositions, checking congruence conditions, and tracking Galois orbits (Galarraga et al., 20 Nov 2025). This method is efficient for moderate degrees, with complexity polynomial in the degree of $F$ once root factorization is available.

5. Degree Sets in Symbolic Dynamics and Graph Theory

In symbolic dynamics, the conceptually analogous computation is for the "degree" of a one-block code $\pi: X\to Y$ between subshifts. For finite-to-one codes, the degree $d_\pi$ equals the minimal fiber size over a transitive point, and is computed by constructing two directed graphs encoding allowable transitions and taking the minimal cardinality of set intersections after reachability computations (Allahbakhshi, 2014).

In graph theory, the degree set of a finite simple graph is the set of distinct vertex degrees. The problem of realizing a minimal size (number of edges) graph with a prescribed degree set $\mathscr D$ is solved by a suite of algebraic-combinatorial algorithms, including the computation of parity adjustments, Erdős–Gallai deficits, and explicit sequence realizations via splitting and aggregation lemmas (Moondra et al., 2020). For interval degree sets and those divisible by their minimal element, exact formulas are provided; in general, the minimal size is computed to within an additive error of $d_0 = \min \mathscr D$.

Counting the number of degree sequences of given properties (e.g., zero-free, connected, biconnected) is accomplished in polynomial time via dynamic programming on partition recurrences (Barnes–Savage) expressing the number of degree sequences and connecting to graphical partition counts (Wang, 2016).

6. Combinatorial Commutative Algebra Methods

For projective varieties, the degree can be realized as the cardinality of a combinatorial set constructed via Gröbner bases, polarization, and Stanley–Reisner (squarefree monomial) decompositions. The algorithm of Stelzer–Yong constructs a finite set of “hieroglyphs” whose cardinality is the degree of the variety. The computation proceeds as follows (Stelzer et al., 2023):

1. Compute a Gröbner basis for the ideal $I$ and extract the initial monomial ideal.
2. Polarize this monomial ideal to squarefree form, preserving degree.
3. Decompose into minimal prime components; each maximal prime corresponds to a hieroglyph (an array encoding the prime generator support).
4. The set of all such hieroglyphs (with minimal support) is $T(V(I),<)$ and $|T| = \deg V(I)$.

While the procedure is doubly-exponential in the number of variables in the worst case (due to Gröbner basis computation), it provides an explicit combinatorial construction of degree sets in many classical cases, with extensions to multidegree, Samuel multiplicity, and specialized families admitting uniform combinatorial interpretations.

7. Complexity, Limitations, and Open Problems

The discussed methods span arithmetic, combinatorial, and algebraic techniques. For curves over Henselian fields with SNC models, degree set computation is efficient and reduces to finite combinatorics (Creutz et al., 2023). For superelliptic covers, the cluster/valuation method is practical for moderate degrees (Galarraga et al., 20 Nov 2025). Combinatorial commutative algebra offers constructive but generally expensive algorithms for degree set and degree computation of varieties (Stelzer et al., 2023). In symbolic dynamics, the degree and class-degree algorithms can be implemented for small alphabets but become intractable as fiber size increases (Allahbakhshi, 2014).

Open challenges include extending these combinatorial methods to singular or non-SNC models, improving complexity bounds in the commutative algebra setting, finding uniform combinatorial objects parametrizing degree sets for broader families, and understanding structural phenomena for degree sets in arithmetic settings beyond the regular and SNC cases.

In summary, the computation of degree sets is now supported by a suite of algorithmic methods that, when applicable, provide a finite and explicit description solely based on combinatorial, geometric, or algebraic data of the object or its model (Creutz et al., 2023, Allahbakhshi, 2014, Stelzer et al., 2023, Galarraga et al., 20 Nov 2025, Moondra et al., 2020, Wang, 2016).