Sometimes in mathematics, an idea is elegant enough that even when it fails, the architecture of its failure is worth documenting. This is the story of an attempt to rescue polynomial differential equations from a rational denominator, only to be stopped by a strict bound on polynomial degrees: the $2m+1$ vs $2m$ regress.

1. The Setup: Probability Distributions and One Marking

In the study of Population Protocols (PP), we often seek to compute target polynomial differential equations, say $\dot{x} = x^2$. To bypass the complexities of stochastic interactions, we operate in the continuous, macroscopic fluid limit. In this setting, the population size is strictly conserved, meaning the total mass of all species always sums to 1. Consequently, the state of the system at any given time is naturally a probability distribution over the available states.

The aesthetic ideal for such a system is to compute complex target probabilities starting from the simplest possible initial distribution: a single species starts with a mass of 1, and all others start at 0.

However, representing a target mathematical variable $x(t)$ within this strict probability distribution is non-trivial. In prior work on computing real numbers with large-population protocols (e.g., DNA 28, 2022), a target quantity had to be “marked” using multiple species—for instance, encoding the value as a fraction relative to a scaling reference species ($x = \frac{Z}{\lambda R}$).

This raises a fundamental open question: Can we achieve “One Marking”? That is, can we use exactly one species to mark a quantity, reading its direct concentration as $[X] = x(t)$?

In a One-Marking system, because the target $x(t)$ cannot be scaled dynamically to absorb mass changes, we are forced to introduce an independent “absorber” species $r$ (along with necessary auxiliary tracking species $z_i$) to ensure the probability distribution always sums to 1:

2. The Rational Impasse

To implement the target dynamics while satisfying mass conservation, we define auxiliary variables that track scaled products of $x$ and $r$ (e.g., $z_1 = \lambda r x$).

By differentiating the conservation equation $\dot{x} + \dot{r} + \sum \dot{z}_i = 0$, we must solve for the dynamics of the absorber, $\dot{r}$.

This is done by applying the chain rule to the auxiliary variables (e.g., the derivative of the first cross term $\dot{z}_1$ is exactly $\lambda(\dot{r}x + r\dot{x})$).

This leads to a fatal structural problem: $\dot{r}$ becomes a rational function (a fraction containing variables in the denominator). For example:

Mass-action kinetics only allows polynomial ODEs (representing molecular collisions). A rational ODE cannot be directly implemented by a CRN. The standard “Method of Unknowns” seems dead in the water.

3. The Bold Idea: Balancing Dilation

Faced with a rational ODE, the instinct is to retreat. But a bold idea emerged: What if we attack the denominator directly?

Let the rational denominator of $\dot{r}$ be $D$. If we willingly alter the target dynamics by multiplying its speed by $D$ (and the required homogenization factor $r$), we get a new “dilated” target dynamics:

When we substitute this new $\dot{x}$ back into the equation for $\dot{r}$, the $D$ in the numerator perfectly cancels the $D$ in the denominator! Suddenly, the entire system—$\dot{x}$, $\dot{r}$, and all $\dot{z}_i$—collapses back into beautiful, clean polynomials. We have successfully “polynomialized” the rational ODE without violating the mass conservation constraint.

4. The Infinite Regress ($2m+1$ vs $2m$)

The polynomials are restored, but are they implementable? We are specifically targeting Population Protocols (PP). A population protocol computes by having finite-state agents interact in pairs (a strictly “two-in, two-out” mechanism). In the macroscopic fluid limit, this physically restricts the system to bimolecular mass-action kinetics. Therefore, our polynomial ODEs can have a maximum degree of exactly 2 in terms of the tracked macroscopic variables.

Suppose we allow our system to track auxiliary variables up to degree $m$ (e.g., if $m=3$, we track $x^3, x^2r$, etc.). Because $D = 1 + \sum \frac{\partial z_i}{\partial r}$, the highest degree in $D$ is $m-1$.

When we apply the Balancing Dilation factor $r \cdot D$ to a standard degree-2 target, we get a new dilated dynamics. Let the original target be denoted as $\dot{x}_{\text{target}} = x^2$.

The degree of our new dynamics $\dot{x}_{\text{new}}$ becomes:

Now, consider the derivative of our highest-degree tracking variable, denoted as $V_{\max} \sim x^m$.

By the chain rule, its derivative $\dot{V}_{\max}$ is proportional to $x^{m-1} \dot{x}_{\text{new}}$. The polynomial degree of this derivative is:

But our CRN only contains tracking variables up to degree $m$. The highest polynomial degree we can form by colliding two of these variables (a bimolecular reaction) is $m + m = \mathbf{2m}$.

5. The Algebraic Obstruction

We have arrived at a clear algebraic obstruction:

No matter how many higher-degree variables we add to the system (increasing $m$), the required polynomial degree for the derivative ($2m+1$) will always exceed the maximum degree our bimolecular reactions can supply ($2m$).

If we track variables up to degree 100, bimolecular combinations can form degree 200, but the chain rule and the Balancing Dilation factor will yield degree 201. The sequence does not terminate.

The Balancing Dilation trick successfully resolved the rational ODE problem, but it introduced a structural degree expansion. This shows that the specific “Method of Unknowns” approach cannot simultaneously satisfy One Marking, strict mass conservation, and bimolecular mass-action kinetics. The polynomial degrees do not close.