Prescribing Infinite Patterns in Arithmetic Dynamics

A linear recurrence can vanish at times $1,2,4,8,16,\ldots$, and at no other times. To see how, work in characteristic two, where $1+1=0$, and consider

$$u(n)=(t+1)^n-t^n-1.$$

Here $n=0,1,2,\ldots$, $t$ is a formal variable, and the terms belong to the field of rational functions $\mathbb F_2(t)$. Each of the three terms is a geometric progression, so their difference satisfies a fixed linear recurrence: every new term is a fixed linear combination of a finite number of previous terms. Repeated squaring gives $(t+1)^{2^a}=t^{2^a}+1$, and a binomial-coefficient argument shows that these powers of two are exactly the positive times when the sequence vanishes.

This classical example, recorded by Lech and discussed in Derksen's paper, illustrates a basic question about arithmetic dynamics. How much infinite behavior can a finite algebraic rule prescribe?

Our manuscript, Realization, Extension, and Orbit Obstructions in Arithmetic Dynamics, gives constructions addressing four questions from papers and preprints dated 2005 to 2019. Two results establish that certain prescribed dynamics can always be realized. Two others identify precise obstructions to proposed universal behavior. The proofs develop three useful techniques: recovering large powers from rational functions, preserving a dynamical system while enlarging its coordinates, and detecting geometric information that pointwise iteration does not reveal.

The oldest question concerns the zero sets of linear recurrences. In a preprint from 2005, published in 2007, Harm Derksen described the possible zero sets in positive characteristic and asked whether every set in his classification could actually occur. This is his Conjecture 3.6, an inverse problem dating back roughly two decades. The general inverse problem was still explicitly described as open in Xie and Yang's May 2026 discussion.

Fix a prime $p$. The permitted sets in characteristic $p$ are called $p$-normal. Their building blocks include arithmetic progressions and sets formed from expressions such as

$$c_0+c_1q^{a_1}+\cdots+c_rq^{a_r}, \qquad q=p^e,$$

where $e\ge1$ is an integer and the exponents $a_i$ vary independently over nonnegative integers. The rational coefficients satisfy integrality conditions, and only nonnegative values of the expression are kept. One may take finite unions and change finitely many individual times. Positive and negative coefficients are both allowed: a pattern built from $3^a-2\cdot3^b$, retaining its nonnegative values, is a typical example of the signed case.

We prove that every $p$-normal set is the exact zero set of a finite-order linear recurrence over the single field $\mathbb F_p(t)$. The recurrence may depend on the set; the field does not. The statement specifies every time, including any prescribed finite changes. It completes the inverse direction of the classification with a uniform choice of coefficient field.

The distinction between knowing the allowed patterns and constructing them matters. A classification gives restrictions on what an existing recurrence can do. An inverse theorem turns those restrictions into a sufficient condition: give us an allowed pattern, and a recurrence exists that realizes exactly that pattern. All the variation can be placed in the coefficients and the recurrence order; one formal variable suffices.

The proof starts with a familiar identity. In characteristic $p$, taking a $p$th power preserves addition and multiplication; this operation is called Frobenius. Over a field with $q$ elements, every element $a$ satisfies $a^q=a$. Introduce a second variable $\xi$, independent of $t$. Evaluating the rational function

$$R(\xi)=\frac{\xi+t^{q^2}}{\xi+t^q}$$

at any $a\in\mathbb F_q$ gives $R(a)=(t+a)^{q^2-q}$. The numerator records a positive power contribution; the denominator records a negative one. Rational functions therefore provide a natural way to retain signed weights in the same object.

The useful new step runs this reasoning backward. Suppose a nonzero rational function over $\mathbb F_q(t)$ has complexity at most $C$, measured by the sum of its reduced numerator and denominator degrees in $\xi$, and $q>4C$. If its values agree with $(t+a)^n$ for an integer $n$ at at least $q-C$ elements $a\in\mathbb F_q$, the reconstruction lemma forces it to be a product of factors $\xi+t^{q^j}$, with nonnegative levels $j$ and signed integer multiplicities. The same multiplicities reconstruct $n$ as a signed sum of powers of $q$.

Why can finitely many evaluations force this structure? Differentiate with respect to $t$ and clear denominators. The resulting polynomial in $\xi$ has bounded degree. Once it vanishes at more points than that degree permits, it must vanish identically. The sampled equalities have become an exact differential identity.

There is a subtlety in characteristic $p$: differentiation sees multiplicities only modulo $p$. A factor occurring $p$ times can disappear from the derivative altogether. The proof retains its full integer multiplicity in the rational function, removes that complete factor, and takes a Frobenius root of the remaining coefficients. Each nonzero removal lowers the degree complexity. Keeping track of the finite-field permutation induced by these roots makes the process consistent across successive stages.

After the last nonzero removal, the remaining integer exponent must be divisible by every power of $p$, forcing it to be zero. The evaluation identities then force the remaining function to be $1$. Reversing the steps recovers the desired powers.

This supplies a finite algebraic test for a pattern involving arbitrarily large exponents. The construction builds on earlier geometric realizations and a decomposition used by Lee and Nam in their 2024 preprint, published in 2025. The signed reconstruction and boundary control allow the full class to be realized by linear recurrences.

To obtain the full zero-set theorem, the proof also controls what happens when parameters collide or move to infinity in a projective closure, which includes limiting parameter configurations. Otherwise, taking a closure could introduce extra zero times. Those boundary arguments retain the original signed weights.

The resulting fixed equations are evaluated along an orbit whose coordinates are geometric progressions. This produces finite sums of exponentials in time, hence linear recurrences. Closure operations on recurrences then assemble the required sets, and a field norm brings temporary finite extensions of the constants back to $\mathbb F_p(t)$ without changing which terms are zero.

The next construction concerns a different kind of preservation. Suppose a dynamical system is defined on a projective variety $X$. Can we place $X$ inside a larger projective space and extend its map to that entire space, while keeping its original action on $X$?

Projective space uses coordinates $[x_0:\cdots:x_N]$, with proportional nonzero tuples representing the same point. A tuple of homogeneous polynomials of the same positive degree defines an everywhere-defined map only when those polynomials have no common nonzero zero, including over extension fields. Coordinate formulas that work perfectly on $X$ can acquire common zeros elsewhere in the ambient space. These are called base points.

For a polarized map, a compatible choice of geometric coordinates has a specified degree $d$. Let $\phi:X\to X$ be the given map. Formally, an ample line bundle $\mathcal L$, a sufficiently high tensor power of which gives a projective embedding of $X$, satisfies $\phi^*\mathcal L\simeq\mathcal L^{\otimes d}$. We treat $d\ge2$. Poonen's 2013 paper asked whether changing the embedding always permits an extension and explicitly left the finite-field case unresolved. Extensions over infinite fields were already available, as were results permitting replacement of the original map by a suitable iterate.

Our result permits a larger ambient dimension and constructs an embedding $j$ and a degree-$d$ map $\Psi$ over the original field such that

$$\Psi\circ j=j\circ\phi.$$

Every original time step is preserved: $\Psi^n\circ j=j\circ\phi^n$ for all $n\ge0$. The embedding $j$ carries every orbit on $X$ to the corresponding $\Psi$-orbit in the ambient space. The theorem works over arbitrary fields, and the construction retains multiplicity information in the equations defining $X$.

The finite-field difficulty has a simple illustration. Remove from the affine line every solution of $x^q-x=0$. The remaining algebraic open set is nonempty, yet it contains no point defined over $\mathbb F_q$. A proof that good coordinate choices form a nonempty open set therefore does not guarantee a choice over the original field.

We choose good coordinates over a finite extension and expand each extension-field coordinate in a basis over the original field. This increases the number of coordinates while preserving polynomial degree. For example, over $\mathbb F_3$, introduce $\alpha$ with $\alpha^2=-1$. Squaring becomes

$$(x+\alpha y)^2=(x^2-y^2)+\alpha(2xy).$$

In the two coordinates $x,y$, this gives the map $[x:y]\mapsto[x^2-y^2:2xy]$. Its coordinate polynomials have no common nonzero zero, even over the algebraic closure, so it is a quadratic map of the whole projective line. In the conjugate coordinates $x+\alpha y$ and $x-\alpha y$, it simply squares each coordinate.

The general proof verifies exactly this geometric property. Finite extensions of finite fields are separable, which makes the change to all conjugate coordinates invertible. Over an algebraic closure, those coordinates separate the expanded map into independent conjugate blocks. Each block has the origin as its only common affine zero, so all the expanded coordinates vanish simultaneously only at the origin. Projectivization therefore introduces no base point. Meanwhile, expanding the section identities on $X$ in the same basis proves the commuting equation above. These identities preserve the original equations and their multiplicities.

Restriction of scalars and separability are classical tools. The contribution here is the homogeneous construction that preserves both the absence of geometric base points and the original dynamical identities. Together with the preparation of the embedding, it gives the extension with degree and time steps unchanged.

The last two results use a much simpler-looking formula:

$$S(u,v)=(u,uv),\qquad S^n(u,v)=(u,u^nv).$$

The first coordinate stays fixed, and the second is repeatedly multiplied by it. Over a finite field, the nonzero elements form a cyclic group. Whether the second coordinate ever reaches $1$ is therefore determined by whether $v$ belongs to the subgroup generated by $u$.

Over $\mathbb F_5$, for instance, taking $u=2$ and $v=3$ makes the second coordinate run through $3,1,2,4,3,\ldots$. Taking $u=4$ and $v=2$ gives $2,3,2,3,\ldots$, which never reaches $1$. Here $3$ belongs to the subgroup generated by $2$, while $2$ lies outside the subgroup $\{1,4\}$ generated by $4$. This subgroup test decides whether the distinguished value is visited.

We build a cubic birational map of the projective plane whose dynamics on a suitable open set have precisely these coordinates. Birational means that the map has a rational inverse on a dense open set. When the multiplicative orbit reaches the distinguished value, the original projective orbit reaches an indeterminacy point. The step into that point is defined; the following step is undefined. The proof checks these boundary transitions in the original map, so counting successful orbits becomes an exact finite-group calculation.

This addresses Conjecture 18.10(b) in the 2018 preprint, published in 2019, of Current Trends and Open Problems in Arithmetic Dynamics. For every dominant rational self-map of projective space, the conjecture predicted that this proportion tends to one over growing extensions of a fixed finite field. Dominance means that the image is algebraically dense, a condition satisfied by our birational map. Weinreich had already questioned that expectation through the pentagram map. Our contribution is an explicit plane map with an exact count and fully determined limiting behavior.

Let $\delta(n)$ be the proportion of points of the projective plane over $\mathbb F_{2^n}$ whose orbit under our map can continue forever. Then

$$\liminf_{n\to\infty}\delta(n)=0, \qquad \limsup_{n\to\infty}\delta(n)=1.$$

Both extremes occur for one fixed map over one fixed base field. Along prime extension degrees, almost every point eventually encounters the obstruction. Along factorial extension degrees, almost every point avoids it.

The source of this oscillation is arithmetic. If a multiplier $u$ has order $h$, exactly $h$ nonzero values of $v$ lie in its subgroup. One of these is $v=1$, which is excluded from the initial projective chart, leaving $h-1$ unsuccessful starting values there. Summing these counts and accounting for the boundary gives the exact number of unsuccessful trajectories. The controlling quantity is the normalized mean element order: the average fraction of the multiplicative group visited by the cycle generated by a multiplier. Larger fractions expose more starting values to the forbidden point.

At prime extension degrees, the new prime factors of the multiplicative group order are forced to be large. At factorial extension degrees, that group order acquires more and more small prime factors. The resulting normalized mean element orders have different limits. The same construction has nonconvergent orbit density over every finite base field, with exact formulas for both extrema.

The geometric set of successful starting points is also Zariski dense over the algebraic closure: no nonzero polynomial equation cuts out a proper algebraic subset containing all of them. This example shows how that geometric abundance can coexist with proportions approaching zero along an infinite sequence of finite extensions. It gives a concrete arithmetic reason that geometric density alone cannot determine finite-field frequency.

The map $S$ also provides the obstruction in the fourth result. Here we construct an everywhere-defined polynomial map in three variables. Its unique fixed point is the origin. A subvariety is geometrically nilpotent if each of its points over the algebraic closure eventually reaches that origin, with a time that may depend on the point. It is nilpotent if a single time works for the whole subvariety. Each individual point belongs to some finite extension field, but no single finite field contains all the points under consideration. Their absorption times can therefore lack a common bound.

Borisov had already constructed examples separating these two properties in a 2015 preprint, published in 2018. His Question 13 asked whether every system has one geometrically nilpotent subvariety from which all the others can be covered by taking forward images and inverse images. Such a subvariety would be a universal seed for this operation.

We construct a system with no such seed, in every prime characteristic. The obstruction comes from an infinite family of surfaces whose ratios on the nonzero coordinate chart lie on the curves

$$C_s:\quad u=v^{p^s},\qquad s=1,2,3,\ldots.$$

At a geometric point, write $(u,v)=(a^{p^s},a)$. The element $a\ne0$ has finite multiplicative order $m$, coprime to $p$. Since $p^s$ is invertible modulo $m$, there is an $n\ge0$ for which $1+np^s\equiv0\pmod m$. Under $S^n$, the second coordinate is then $1$. The original three-dimensional map turns this event into absorption at the origin. Its remaining boundary points also reach the origin and never return to the nonzero chart. Thus every point of each surface is eventually absorbed, and any orbit still in that chart can be followed in ratio coordinates throughout its history.

The curves nevertheless retain different algebraic information. Set $K=\overline{\mathbb F_p}$, the algebraic closure of $\mathbb F_p$, and now parametrize the whole curve by an indeterminate $\tau$. Its field of rational functions is $K(\tau)$, while the functions coming from the retained coordinate $u$ form the subfield $K(\tau^{p^s})$. The extension

$$K(\tau^{p^s})\subset K(\tau)$$

has degree $p^s$. Recovering $\tau$ from $u$ requires adjoining a $p^s$-th root; this is called a purely inseparable extension. That degree survives every forward or backward iterate of $S$, because $S$ fixes $u$.

This gives a particularly concrete distinction. On every finite field of characteristic $p$, raising elements to a $p^s$-th power is a bijection. On $K(\tau)$, its image is $K(\tau^{p^s})$, and the full field has degree $p^s$ over that image. In characteristic two, the curves $u=v^2$ and $u=v^4$ are abstractly isomorphic, but their projections to the same $u$-axis have degrees two and four. Retaining that projection supplies an invariant that separates their dynamics.

An infinite list of degrees alone would not exclude every possible seed. The proof also shows that the Zariski closure of a geometrically nilpotent seed's ratio projection has dimension at most one: a two-dimensional projection would contain points that never reach the origin. There are only finitely many irreducible curve components in such a projection. Each component can account for at most one of the degrees $p^s$, even after arbitrary forward and backward iterates. It follows that a fixed seed can cover only finitely many members of our family. The argument continues to work when its forward images are enlarged to their Zariski closures, the smallest closed algebraic sets containing them.

The manuscript is accompanied by Lean 4 source and a detailed account of its formalization scope. The inverse recurrence theorem, the orbit-count and density results in a projective-point model, and the core nonexistence theorem in its algebraic-closure-point formulation have complete formal proofs from their stated inputs. For projective extension, the coordinate construction and the final projective diagram are formalized; the passage from an arbitrary polarized scheme to those coordinate inputs remains to be formalized. The paper identifies this gap and the additional ancillary statements outside the verified scope.

The resulting tools have concrete uses: constructing prescribed zero sets, embedding a dynamical system without changing its iteration, computing exact orbit frequencies, and proving that a proposed universal construction cannot exist. Each comes with an explicit mechanism and a precise account of the information it must preserve.