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A Positive Proportion of the Reduced D'Arcais Polynomials is not Hurwitz

This paper disproves a conjecture by the second and third authors by demonstrating that a positive proportion of the reduced D'Arcais polynomials are not Hurwitz polynomials.

Original authors: Steven Charlton, Bernhard Heim, Markus Neuhauser, Johann Stumpenhusen, Robert Tröger

Published 2026-08-20

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CC BY 4.0

Original authors: Steven Charlton, Bernhard Heim, Markus Neuhauser, Johann Stumpenhusen, Robert Tröger

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

In the vast landscape of mathematics, there are objects that behave like well-behaved citizens, following predictable rules that keep them stable and orderly. Among these are special polynomials, which are algebraic expressions built from numbers and variables. For decades, mathematicians have been fascinated by a particular family of these expressions known as D'Arcais polynomials. These are not just random collections of numbers; they are deeply connected to the partition function, a fundamental concept that counts the number of ways a whole number can be broken down into a sum of smaller whole numbers. This connection links them to the Dedekind eta function, a sophisticated tool used to study the symmetries of shapes in higher dimensions. The central question for a long time was whether these polynomials always possess a specific kind of stability. In the language of mathematics, a polynomial is called "Hurwitz" if all its solutions, or roots, lie in a specific region of the complex number plane that guarantees stability. If a polynomial is Hurwitz, it behaves predictably; if it is not, it can exhibit chaotic or unstable behavior. For a long time, it was believed that these D'Arcais polynomials were always Hurwitz, except for a single trivial exception at the origin. This belief was so strong that it was supported by checking the first thousand examples, all of which appeared to follow the rule.

However, a new study has overturned this long-held assumption. The researchers, a team of mathematicians, have proven that the belief is false. They demonstrated that a significant, non-zero portion of these polynomials are not Hurwitz. In other words, there is a positive proportion of natural numbers for which the corresponding D'Arcais polynomial fails the stability test. The team did not just find a single oddball example; they showed that these failures are common enough to appear frequently as you look at larger and larger numbers. Their work provides a definitive answer to a conjecture that had stood for some time, showing that the pattern of stability is not universal, even though the first instance of this breakdown occurs at an astronomically large number far beyond current computational reach.

To reach this conclusion, the researchers had to look deeper than simply calculating the roots of the polynomials, a task that would be impossible for the massive numbers involved. Instead, they used a set of mathematical criteria known as the Hurwitz-Routh test. This test acts like a checklist of conditions that a polynomial must satisfy to be considered stable. If any single condition on the list is violated, the polynomial is unstable. The team focused on a specific condition involving the coefficients of the polynomial—the numbers that multiply the variables. They derived precise estimates for how large or small these coefficients could be. By comparing the lower bounds of some coefficients against the upper bounds of others, they constructed a scenario where the stability condition must fail.

The proof relies on the behavior of a specific function related to the sum of divisors of a number. The researchers showed that for certain very large numbers, the relationship between the coefficients forces the stability condition to break. They calculated a specific threshold number, a factorial of a massive integer, beyond which the failure is guaranteed to occur. This threshold is an unimaginably large number, far exceeding the number of atoms in the observable universe. Yet, the mathematical logic holds firm: once you pass this point, the polynomials are no longer Hurwitz. Furthermore, because the properties they used repeat for multiples of this number, the failure is not a one-time event but a recurring phenomenon that happens with regularity. It is important to note that their specific method does not allow them to identify the absolute smallest number for which the polynomial is not Hurwitz; they have only proven that such numbers exist and occur with positive frequency.

The study also sheds light on why this was so hard to detect earlier. The first thousand examples checked by previous researchers were simply too small to reveal the instability. The numbers required to trigger the failure are so vast that they lie far beyond the reach of direct computation or standard numerical observation. The researchers had to rely on theoretical bounds and asymptotic estimates to prove that the instability exists, rather than finding a specific counterexample by brute force. They noted that if a famous unsolved problem in mathematics known as the Riemann Hypothesis is true, the threshold for this failure is even higher, pushing the first guaranteed example even further out into the realm of the incomprehensibly large.

Despite proving that these polynomials are not always stable, the researchers leave several questions open for the future. They do not know exactly where the very first unstable polynomial appears, only that it must exist before their calculated threshold. They also wonder about the distribution of the roots that cause this instability. Visualizations of the roots for smaller numbers show them clustering in specific patterns, and the researchers speculate that these patterns might eventually cross into the unstable region. They also ask whether there are infinitely many distinct unstable roots and if the polynomials are always stable for prime numbers. The work stands as a correction to a long-standing intuition, showing that even in the rigid world of number theory, patterns that seem universal can break down when pushed to the extremes.

Technical Summary: A Positive Proportion of the Reduced D'Arcais Polynomials is Not Hurwitz

Problem Statement
The paper addresses a conjecture regarding the analytic properties of D'Arcais polynomials, denoted as $P_n^\sigma(X)$ , which are defined via the generating function of the coefficients of powers of the Dedekind $\eta$ -function. Specifically, the second and third authors previously conjectured that the reduced D'Arcais polynomials, $R_n^\sigma(X) := P_n^\sigma(X)/X$ , are Hurwitz polynomials for all natural numbers $n$ . A polynomial is defined as Hurwitz if all its zeros lie strictly within the open left complex half-plane. This conjecture was motivated by empirical observations of the first 1,000 polynomials, where all non-real roots appeared to reside in the left half-plane. The central question is whether this property holds universally or if counterexamples exist.

Methodology
The authors employ the Hurwitz–Routh criterion to determine the stability of the polynomials. For a polynomial $f(X) = \sum a_m X^m$ , the criterion states that $f$ is Hurwitz if and only if a sequence of determinants $\Delta_k(f)$ are strictly positive for all $1 \le k \le d$ . The authors focus on the reciprocal polynomial $f(X) = X^{n-1}R_n^\sigma(X^{-1})$ , noting that $R_n^\sigma(X)$ is Hurwitz if and only if $f(X)$ is.

The proof strategy involves analyzing the second Hurwitz determinant, $\Delta_2(f)$ . For the specific structure of the D'Arcais polynomials, $\Delta_2(f)$ is expressed in terms of the coefficients $p_n^\sigma(k)$ :
$\Delta_2(f) = p_n^\sigma(2)p_n^\sigma(3) - p_n^\sigma(1)p_n^\sigma(4)$
If $\Delta_2(f) < 0$ , the polynomial is not Hurwitz. The authors aim to show that for a positive proportion of $n$ , the inequality $p_n^\sigma(1)p_n^\sigma(4) > p_n^\sigma(2)p_n^\sigma(3)$ holds.

To establish this, the authors utilize:

Lower Bounds: They derive a lower bound for $p_n^\sigma(4)$ using combinatorial arguments involving compositions of $n$ , and a lower bound for $p_n^\sigma(1)$ based on harmonic numbers $H_\ell$ when $\rho(\ell)$ (the least common multiple of $1, \dots, \ell$ ) divides $n$ .
Upper Bounds: They utilize previously established upper bounds for $p_n^\sigma(2)$ and $p_n^\sigma(3)$ from their prior work [4], simplified using estimates for the generalized sum-of-divisors function $\sigma_{-k}(n)$ and logarithmic terms.
Asymptotic Analysis: They compare the growth rates of the product of the lower-bounded terms against the product of the upper-bounded terms.

Key Contributions and Results
The primary result is Theorem 1.2, which asserts that the set of natural numbers $n$ for which $R_n^\sigma(X)$ is not Hurwitz has a positive lower asymptotic density. Specifically, if $A(n)$ counts the number of such counterexamples up to $n$ , then:
$\liminf_{n \to \infty} \frac{A(n)}{n} > 0$

The proof demonstrates that the condition for non-Hurwitz behavior is satisfied whenever:
$\sigma_{-1}(n) > \frac{25,046,441}{116,352} \approx 215.264$
Since $\sigma_{-1}(n)$ is unbounded, such $n$ exist. The authors explicitly construct a counterexample by choosing $n = \ell!$ , where $\ell$ is the ceiling of $\exp(25,046,441/116,352)$ . This yields a specific integer $\ell \approx 3.07 \times 10^{90}$ , and consequently $n = \ell!$ is a number with approximately $10^{95}$ digits. The authors note that due to the multiplicative property of $\sigma_{-1}$ , all multiples of this factorial also serve as counterexamples, ensuring a positive proportion of integers fail the Hurwitz condition.

Significance and Claims
The paper provides a definitive negative answer to Conjecture 1.1, showing that the D'Arcais polynomials are not universally Hurwitz. The authors emphasize several nuances regarding their findings:

Existence vs. Minimality: The proof relies on the sufficient condition $\Delta_2(f) < 0$ . The authors explicitly state that this is not a necessary condition for a polynomial to be non-Hurwitz. Therefore, their approach does not identify the smallest $n$ for which the conjecture fails; the actual first counterexample could be significantly smaller.
Magnitude of Counterexamples: The explicit counterexample derived ( $n = \ell!$ ) is astronomically large, exceeding the number of atoms in the observable universe. The authors attribute this to the "sheer size of our error terms" in the bounds.
Relation to the Riemann Hypothesis: Under the assumption of the Riemann Hypothesis, Robin's theorem implies $\sigma_{-1}(n) < e^\gamma \log(\log n)$ . The authors note that under this hypothesis, the inequality required for their counterexample would not hold for any $n < 10^{1052}$ , further highlighting the vast gap between their theoretical bound and potential smaller counterexamples.
Open Questions: The paper concludes by listing open challenges, including whether there are infinitely many distinct roots in the positive half-plane, the behavior of the root distribution trajectories (as seen in Figure 1), and whether $R_p^\sigma(X)$ is Hurwitz for all prime numbers $p$ .

In summary, the paper establishes that the Hurwitz property for D'Arcais polynomials fails for a positive proportion of natural numbers, refuting the previous conjecture, though the specific counterexamples identified are of immense magnitude due to the limitations of the analytic bounds employed.

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