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On the generating series of the degree sequence

This paper investigates the generating series of degree sequences for monomial self-maps on projective toric varieties, establishing conditions for their transcendence and non-holonomicity while proving that, for toric surfaces, the series' reduction modulo $p$ is transcendental for all but finitely many primes.

Original authors: Quang-Khai Nguyen

Published 2026-08-04

📖 8 min read🧠 Deep dive

CC BY 4.0

Original authors: Quang-Khai Nguyen

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are watching a cosmic dance where shapes twist, stretch, and fold over and over again. In the world of algebraic dynamics, mathematicians study these dances by tracking how "complex" a shape gets with every step. They use a special ruler called a "degree" to measure this complexity. Sometimes, these dances are predictable, like a clock ticking in a steady rhythm. Other times, they are chaotic, like a storm that never repeats itself. The big question is: Can we predict the future steps of these dances just by looking at the past? To answer this, mathematicians use a powerful tool called a "generating series." Think of this series as a magical recipe book. If you feed it the sequence of complexity numbers from the dance, it spits out a single, infinite mathematical formula. The shape of this formula tells us everything about the dance: is it simple and orderly, or is it wild and unpredictable?

This paper, written by Quang-Khai Nguyen, dives deep into a specific type of dance performed on geometric shapes called "toric varieties." These are fancy, multi-dimensional shapes built from grids and cones, often used to model complex systems in physics and math. The author focuses on "monomial maps," which are dances where the movement is defined by simple multiplication rules (like squaring a number or multiplying two variables). The goal is to understand the "generating series" for the complexity of these dances. The paper asks: Is the recipe book for these dances a simple, rational fraction (like 1/2), or is it a wild, transcendental beast that cannot be tamed by simple equations? The answer turns out to be a fascinating mix of order and chaos, revealing that even in these seemingly simple geometric worlds, the underlying patterns can be incredibly complex and impossible to predict perfectly.

The Dance of Complexity

The story begins with a simple idea: tracking how a shape changes. Imagine you have a piece of clay (a geometric shape) and you keep squishing and stretching it. Every time you do this, the shape gets more complicated. Mathematicians count this complication using "degrees." If you stretch it once, it might have a degree of 2; twice, maybe 4; and so on. This list of numbers is the "degree sequence."

The author studies what happens when you take this list of numbers and turn it into a "generating series." You can think of this series as a magical machine. You feed it the number 1, then the number 2, then 4, then 8, and it spits out a single, infinite mathematical expression. The behavior of this expression tells us the secret nature of the dance.

If the expression is a "rational function," it's like a song with a repeating chorus. It's predictable, orderly, and easy to understand. If the expression is "transcendental," it's like a song that never repeats, with a melody that spirals out into infinity in a way that no simple formula can capture. The paper proves that for certain types of dances on these toric shapes, the generating series is definitely transcendental. It's not just a little bit complex; it's so complex that it has a "natural boundary."

The Wall of Chaos

Here is the most exciting part of the discovery. The author shows that for these specific dances, the mathematical formula hits a "wall" that it cannot cross. Imagine you are walking along a path that gets more and more interesting. Suddenly, you reach a circle. On the inside of the circle, everything is smooth and predictable. But the moment you try to step outside that circle, the path dissolves into a jagged, chaotic mess. You cannot walk further.

In math terms, this circle is called the "circle of convergence." The paper proves that for these monomial maps, the circle of convergence is a "natural boundary." This means the formula is so wild that it cannot be extended even a tiny bit beyond that circle. It's like a cliff edge. Because of this, the formula is "transcendental" and "non-holonomic." In plain English, this means the dance is fundamentally unpredictable in a very deep way. You can't write down a simple equation to describe the whole pattern, and you can't even write a simple differential equation (a rule for how the pattern changes) to capture it.

The author establishes specific conditions for when this happens. It occurs when the "eigenvalues" (the secret numbers that control the speed and direction of the dance) have a particular relationship. Specifically, if two of these numbers are complex conjugates (like mirror images in the complex plane) and their ratio is not a simple fraction of a circle (not a "root of unity"), then the dance hits that chaotic wall. The paper proves this rigorously, showing that the complexity of the degree sequence is not just a fluke, but a guaranteed feature of these specific geometric dances.

The Surface Case: A Complete Picture

The paper gets even more interesting when it looks at two-dimensional surfaces (like a sphere or a donut shape made of cones). Here, the author provides a complete "either/or" answer. It's a perfect dichotomy:

The Orderly Case: If the dance parameters are "nice" (like if the eigenvalues are real numbers or if they repeat in a simple cycle), then the generating series is a rational function. It's a simple, predictable song.
The Chaotic Case: If the parameters are "wild" (complex conjugates with a non-repeating ratio), then the series has a natural boundary. It's a wild, unpredictable storm.

There is no middle ground. The paper proves that these are the only two possibilities. This is a big deal because it settles a long-standing question about whether these series could be something in between. The answer is a firm "no."

The Secret Code of Primes

One of the most playful and surprising parts of the paper involves "reducing modulo p." This is a mathematical game where you take all the numbers in your sequence and divide them by a prime number (like 2, 3, 5, 7), keeping only the remainder. It's like looking at the dance through a kaleidoscope that only shows you colors in a specific palette.

The author answers a question posed by another mathematician named Bell. The question was: If you play this game with a prime number, does the resulting pattern look simple (algebraic) or complex (transcendental)? The paper proves that for almost all prime numbers, the pattern remains complex and transcendental. It's as if the wildness of the dance is so deep that even when you squint through the lens of a prime number, the chaos still shines through. This is a strong result, showing that the complexity isn't just an illusion of the big numbers; it's baked into the very structure of the sequence.

The Twin Dance Mystery

Finally, the paper tackles a mystery about "rigidity." Imagine you have two different dancers, $\phi$ and $\phi'$ , performing on the same stage. You watch them, and you notice that their complexity scores (the degree sequences) are exactly the same. Does this mean they are doing the exact same dance?

The paper says: Not exactly, but they are very close cousins. If their complexity scores match, then after a few steps (specifically, after 1, 2, 3, 4, or 6 repetitions of the dance), they will be "semi-conjugate." This is a fancy way of saying they are related by a simple transformation. They aren't identical twins, but they are part of the same family. The author proves that if the complexity sequences match, the underlying rules of the dance must be linked in a very specific, rigid way. This is a powerful result because it shows that the complexity sequence is a very strong fingerprint; it almost uniquely identifies the dance.

Why This Matters

Why should a curious teenager care about this? Because it shows us that even in the most structured, mathematical worlds, there is room for true, unbreakable chaos. We often think that if we have a simple rule (like a monomial map), the result must be simple. This paper proves that simple rules can generate infinitely complex patterns that hit a wall of unpredictability. It's a reminder that in mathematics, as in life, things that look simple on the surface can hide deep, wild secrets that we can measure but never fully tame. The paper doesn't just say "it's complicated"; it draws a precise map of where the order ends and the chaos begins, proving that for these specific geometric dances, the chaos is absolute and unyielding.

Technical Summary: On the generating series of the degree sequence

Problem Statement
This paper investigates the analytic and arithmetic properties of the generating series associated with the degree sequences of dominant monomial self-maps on projective toric varieties. While the asymptotic behavior of these sequences (the dynamical degrees) is well-understood and known to be algebraic in specific settings, the finer structure of the sequences themselves remains less explored. The central problem is to determine the nature of the generating power series $\Delta_{k,\phi,D}(z) = \sum_{n \ge 0} \deg_{D,k}(\phi^n) z^n$ within the hierarchy of power series: Rational ( $\text{Rat}$ ) $\subset$ Algebraic ( $\text{Alg}$ ) $\subset$ Holonomic ( $\text{Hol}$ ) $\subset$ Differentially Algebraic ( $\text{DiA}$ ) $\subset$ Formal Power Series ( $\mathbb{C}[[z]]$ ). Specifically, the author seeks to establish conditions under which these series are transcendental (not algebraic) and non-holonomic, and to analyze their singularities.

Methodology
The approach combines tools from toric geometry, spectral theory, and the theory of piecewise linear functions.

Geometric Interpretation: The degree sequence $\deg_{D,k}(\phi^n)$ is expressed using mixed volumes of polytopes associated with the divisor $D$ and the map $\phi$ . Following established dictionary results between intersection theory and mixed volumes, the degree is shown to be the value of a piecewise linear function $h$ evaluated at the sequence of powers of an auxiliary matrix (specifically, the $k$ -th exterior power of the matrix $A$ defining the monomial map).
Spectral Reduction: Under specific spectral assumptions on the matrix $A$ (ordering of eigenvalues and the presence of complex conjugate pairs with arguments incommensurable with $2\pi$ ), the problem is reduced to a two-dimensional setting. The dominant part of the series is isolated, where the auxiliary matrix acts conformally (as a multiplication by a complex number $\chi$ ).
Fourier Analysis of Piecewise Linear Functions: The core analytic argument involves studying series of the form $\sum h(\chi^n) z^n$ . By identifying the plane with $\mathbb{C}$ , the piecewise linear function $h$ is restricted to the unit circle. Its Fourier series is analyzed; crucially, the discontinuity of the derivative of $h$ implies that infinitely many Fourier coefficients are non-zero.
Arithmetic and Recurrence Properties: The Skolem–Mahler–Lech theorem is employed to show that the non-zero Fourier coefficients occur along arithmetic progressions. This density of singularities is used to prove the existence of a natural boundary. For the arithmetic results (reduction modulo $p$ ), the paper utilizes Christol's theorem from automata theory, linking algebraicity over finite fields to the weak periodicity of coefficient sequences.

Key Contributions and Results

Theorem A (Natural Boundary and Transcendence): For a dominant monomial self-map $\phi$ on a projective toric variety of dimension $d \ge 2$ , if the eigenvalues of the associated matrix $A$ satisfy specific gap conditions ( $|\mu_{k-1}| > |\mu_k| = |\mu_{k+1}| > |\mu_{k+2}|$ ) and the ratio of the dominant complex conjugate eigenvalues is not a root of unity, then the generating series $\Delta_{k,\phi,D}(z)$ has the circle of convergence $C(0, \lambda_k^{-1})$ as a natural boundary. Consequently, the series is transcendental and non-holonomic over $\mathbb{C}(z)$ .
Corollary B (Application to Rational Maps): The result is applied to a specific class of rational maps $f_\mu = g \circ \phi_\mu$ on $\mathbb{P}^2$ (where $g$ is a birational involution). It is shown that the generating series for these maps is meromorphic in a specific disk with a simple pole but admits a natural boundary, extending previous results on transcendental dynamical degrees.
Corollary C (Algebraic Independence): The paper establishes that the transcendence degree of the field generated by the set of all such generating series over $\mathbb{C}(z)$ is infinite. Specifically, for any monomial map satisfying the conditions of Theorem A, the series associated with its iterates $\phi^\ell$ are algebraically independent over $\mathbb{C}(z)$ .
Theorem D (Sharp Dichotomy for Surfaces): In the case of toric surfaces ( $d=2$ $d = 2$ ), the results are sharp. The paper proves a dichotomy:
1. If the eigenvalues are complex conjugates and their ratio is not a root of unity, the series has a natural boundary and is transcendental. Furthermore, its reduction modulo $p$ is transcendental over $\mathbb{F}_p(z)$ for all but finitely many primes $p$ .
2. Otherwise, the series is a rational function.
  This answers a question posed by Bell regarding the transcendence of reductions modulo $p$ .
Theorem E (Rigidity): A rigidity result is proven for absolutely irreducible monomial surface maps. If two such maps $\phi$ and $\phi'$ share the same degree sequence, they are semi-conjugate up to a power $u \in \{1, 2, 3, 4, 6\}$ .

Significance and Claims
The paper claims that even in relatively simple dynamical systems (monomial maps on toric varieties) where dynamical degrees are explicitly computable and algebraic, the associated degree sequences can exhibit significant complexity. The existence of a natural boundary for the generating series demonstrates that these sequences do not satisfy linear recurrence relations with polynomial coefficients (non-holonomic) and are not algebraic.

The results provide a generalization of previous findings on rationality and natural boundaries in dynamical systems (citing works such as [27], [7], [30], [38]). By establishing the transcendence of reductions modulo $p$ for almost all primes, the paper connects the geometric dynamics of toric varieties to arithmetic properties of power series, utilizing deep results from automata theory (Christol's theorem). The rigidity theorem (Theorem E) further highlights the structural constraints imposed by the degree sequence on the underlying monomial maps.

The author explicitly notes that while the series are proven to be transcendental and non-holonomic, the question of whether they are differentially transcendental (hypertranscendental) remains open, though partial progress is made in Remark 5.3 for a related series. The paper does not claim to resolve the general behavior of dynamical degrees for arbitrary rational maps, but rather focuses on the monomial case to reveal intrinsic complexity in the generating series.