First Appearance Degree Explained

• First appearance degree is the minimal index or power at which a system exhibits a novel property, serving as a key invariant in algebra, combinatorics, and number theory.
• It is applied to measure the onset of nontrivial syzygies in polynomial systems, divisibility in Lucas sequences, and symmetry break in matrix analyses.
• Insights from its explicit formulas and bounds help optimize algorithms and reveal deep structural organization in enumerative and algebraic frameworks.

The first appearance degree is a fundamental notion across algebra, combinatorics, computational number theory, and enumerative structures, capturing the earliest instance at which a prescribed phenomenon, algebraic failure, or combinatorial novelty occurs in a sequence or system. The concept appears under varying terminologies such as "first fall degree," "order of appearance," or "rank of apparition," and serves as a critical invariant controlling algorithmic or structural thresholds in polynomial systems, recurrence sequences, matrix statistics, and rational enumerations. Below is a comprehensive exposition of first appearance degree, with precise definitions, canonical constructions, and recent explicit formulas from multiple research domains.

1. Definitions and Formal Frameworks

The first appearance degree delineates the least index, power, or degree at which a nontrivial event—such as a failure of injectivity, vanishing symmetry, or first occurrence of a divisor—is provably realized. Several settings exemplify this principle:

Polynomial Systems (First Fall Degree):

Given an ideal $$I\subset S = F[X_0,\ldots,X_{N-1}]/(X_0^2,\ldots,X_{N-1}^2)$$ generated by homogeneous polynomials $h_1,\ldots,h_r$ of degree $d$, the first fall degree $D_\mathrm{ff}$ is

$$D_\mathrm{ff} = \min \left\{ j \geq d : \dim_F(I \cap S_j) < \dim_F(S^r_{j-d} / U_{j-d}) \right\},$$

where $U_{j-d}$ encodes the obvious syzygies. The induced map $$\bar\varphi_{j-d}: S^r_{j-d}/U_{j-d} \to I \cap S_j$$ fails to be injective at the first fall degree (Kousidis et al., 2019).

Lucas Sequences (Order of Appearance):

For the first Lucas sequence $(U_n)$ with $\gcd(m,b) = 1$, the order of appearance (also "first-appearance degree") is

$\tau(m) := \min\{k \geq 1 : m \mid U_k\},$

the smallest index $k$ such that $m$ divides $U_k$ (Xiao et al., 25 Feb 2025).

Matrix Statistics (Alternating Power Difference):

For $f_A(\sigma) = \operatorname{tr}(AP_\sigma)$ and its alternating power difference $\mathrm{APD}_m(f_A)$, the first appearance degree $m_1(f_A)$ satisfies

$$m_1(f_A) = \min\{m \geq 1 : \mathrm{APD}_m(f_A) \neq 0\},$$

marking the breakdown of even/odd power-sum symmetry (Takemura, 20 Dec 2025).

Calkin–Wilf Enumeration (First-Appearance Map):

For the breadth-first enumeration $\rho_i = a/c$ of $\mathbb{Q}_{>0}$, the map

$$\pi(d) = \min\{i \geq 0 : \mathrm{den}(\rho_i) = d\}$$

gives the earliest occurrence of denominator $d$—termed its first-appearance degree (Bilokon, 29 Aug 2025).

2. First Appearance Degree in Polynomial and Syzygy Systems

The first fall degree, central to Gröbner basis analysis, encapsulates the degree at which the module of syzygies for a system of homogeneous polynomials develops nontrivial relations outside the expected (trivial) ones. In the context of binary extension fields $\mathbb{F}_{2^n}$, systems derived by Weil descent from Semaev's summation polynomials for the index-calculus attack on the elliptic curve discrete logarithm problem (ECDLP) provide a canonical setting.

If $s_0,\dots,s_{n-1}$ is the Weil-descended system from the $(m+1)$-th Semaev polynomial $S_{m+1}(x_1,\ldots,x_{m+1})$, then for $n'\ge m\ge 3$, the improved bound for first fall degree is

$D_\mathrm{ff} \le m^2 - m + 1,$

significantly sharper than the previously known $D_\mathrm{ff} \le m^2 + 1$. For $m=2$, sharpness at $D_\mathrm{ff} = 2$ is established. This degree governs the onset of new syzygies and strongly influences the degree of regularity, which in turn controls algorithmic complexity in F4/F5 Gröbner bases (Kousidis et al., 2019).

3. Order of Appearance and Rank of Apparition in Linear Recursions

The order of appearance, or first-appearance degree, in linear recurrence sequences such as Lucas or Fibonacci numbers, quantifies the smallest index $k$ where a given number $m$ divides a member of the sequence. Key properties emerged from divisibility-index lemmas, GCD-index relations, and $p$-adic valuations.

Explicit formulas for products in Lucas sequences—e.g., $\tau(U_m V_n)$, $\tau(U_m U_n)$—are derived as functions of $\gcd(m,n)$, least common multiples, and subsidiary recurrence values. For example: $\tau(U_m U_n) = [m,n]\,U_{\gcd(m,n)},$

$$\tau(U_m V_n) = \begin{cases} 2[m,n], & \nu_2(m)\le \nu_2(n),\ [m,n]\,V_d, & \nu_2(m) > \nu_2(n), \end{cases}$$

where $d = \gcd(m, n)$ and $[m,n]$ is the least common multiple (Xiao et al., 25 Feb 2025).

4. First Appearance Degree in Matrix Theory and the Alternating Power Difference

In matrix analysis, the first appearance degree $m_1(f_A)$ for the trace-function $f_A(\sigma)$ characterizes the minimal positive integer for which the alternating sum of $m$-th powers of $f_A$ over even and odd permutations becomes nonzero. For families such as the identity $I_n$, Hilbert $H_n$, Vandermonde $V_n$, circulant $C_n$, and Pascal $P_n$ matrices, closed-form formulas are conjectured and in many cases numerically confirmed:

Matrix Family	First Appearance Degree $m_1$	First Nonzero Value of APD
Identity $I_n$	$n-1$	$n!$
Circulant $C_n$	$n-1$	$(-1)^{T_{n-1}} n^{n-2} n!$
Hilbert $H_n$	$n-1$	$\det(H_n) \cdot n \cdot n!$
Vandermonde $V_n$	$n-1$	$(n{-}1)! \prod_{k=1}^{n-1}k!$
Pascal $P_n$	$n-1$	$(n-1)!$
Mult. Table $M_n$	$T_{n-1}$ $(=\frac{n(n-1)}2)$	$T_{n-1}! \prod_{k=1}^{n-1}k!$

For lattices with exceptional symmetry, $m_1(A)=\infty$. For others, the vanishing of $\mathrm{APD}_m(f_A)$ for $1\le m < m_1$ expresses Prouhet–Tarry–Escott-type power-sum equalities, and the abrupt failure at $m_1$ highlights deep interplay between combinatorial invariants and algebraic structure (Takemura, 20 Dec 2025).

5. First Appearance Phenomena in Rational Enumerations

In the context of the Calkin–Wilf enumeration of positive rationals, the first appearance map $\pi(d)$ records, for each denominator $d$, the earliest index in the breadth-first traversal where a reduced fraction of that denominator appears. The row-start index $i_0(a)$ organizes fractions by denominator, and a level $a$ is said to "lock" if $\pi(a)=i_0(a)$.

A central result—termed the pairing or simultaneous novelty theorem—asserts that for every $n\ge2$, there exists $i\in\{0,\ldots,n-2\}$ such that denominators $n-i$ and $n+i$ lock symmetrically around $i_0(n)$: $\pi(n\pm i)=i_0(n)\pm i.$ This combinatorial symmetry, supported by local coherence and discrete intermediate-value theorem arguments, has implications for efficient computation of $\pi(d)$ and illuminates the underlying group-theoretic organization in the enumeration via the free monoid representation in $SL_2(\mathbb{Z})$ (Bilokon, 29 Aug 2025).

6. Algorithmic Repercussions and Structural Implications

The first appearance degree typically controls the complexity threshold for algebraic algorithms and the abrupt onset of combinatorial asymmetry. In Gröbner basis computations for ECDLP, lowering the first fall degree from $m^2+1$ to $m^2-m+1$ reduces the predicted crossover for subexponential attacks by hundreds of field-extension degrees, refining the subexponential scaling regime.

In recurrence analysis, explicit order-of-appearance formulas extend classical results to wide families of Lucas-type sequences, allowing effective computation and finer control over periodicity and divisibility phenomena. In matrix theory, the first appearance degree provides a new invariant for measuring matrix symmetry, even/odd moment balance, and the emergence of group-theoretic features, with closed-form expressions often involving superfactorials and determinants.

In summary, the first appearance degree unifies multiple domains as a sensitive, algebraically meaningful threshold at which qualitative or computational features of a system are first realized. Its explicit determination yields deep combinatorial, algebraic, and analytic insights, and continues to fuel developments at the intersection of algebraic geometry, number theory, combinatorics, and computational mathematics.