---
title: 'Bounded Skolem Problem: Complexity & Algorithms'
url: https://www.emergentmind.com/topics/bounded-skolem-problem
type: topic
---

# Bounded Skolem Problem: Complexity & Algorithms

Searching arXiv for recent and relevant papers on the bounded Skolem problem and closely related variants.
The bounded Skolem problem is a family of decision problems about whether a linear recurrence sequence or, in a continuous analogue, a solution of a linear differential equation attains the value \(0\) within a prescribed bounded domain. In the discrete setting, one standard formulation takes as input an integer linear recurrence sequence \(\langle u_n\rangle_{n=0}^\infty\) and a bound \(N\in\mathbb N\) written in binary, and asks whether there exists \(n\in\{0,\ldots,N\}\) such that \(u_n=0\). A distinct specialization asks for zeros only at indices of the form \(n=\ell p^k\) with \(\ell,k\le c\) and \(p\) prime. In the continuous setting, the bounded problem asks whether the unique real-valued solution of a linear differential equation has a zero in a rational interval \([a,b]\) [2507.11234] [2006.07432] [1506.00695].

## 1. Formal setting and principal variants

The classical Skolem Problem asks, for a linear recurrence sequence \((u_n)\), whether there exists any \(n\in\mathbb N\) such that \(u_n=0\). For an integer LRS of order \(d\), the recurrence has the form
\[
u_{n+d}=a_{d-1}u_{n+d-1}+\cdots+a_0u_n,
\]
with integer coefficients \(a_0,\ldots,a_{d-1}\), and the minimal such \(d\) is called the order of the LRS. If the characteristic polynomial
\[
g(X)=X^d-a_{d-1}X^{d-1}-\cdots-a_0
\]
has distinct algebraic roots \(\alpha_1,\ldots,\alpha_r\) with multiplicities \(m_1,\ldots,m_r\), then the closed form is
\[
u_n=\sum_{i=1}^r P_i(n)\,\alpha_i^n,
\]
where each \(P_i(n)\) has degree \(m_i-1\) [2507.11234].

The bounded discrete problem restricts the search space to a finite initial segment. Its computational content is nontrivial because the bound \(N\) is part of the input in binary, so the interval \([0,N]\) may be exponentially large in the input length. The 2025 low-order complexity work explicitly studies this version and shows that, for every fixed \(d\), there is a randomised polynomial-time algorithm on LRS of order at most \(d\) [2507.11234].

A different bounded variant, studied by Kenison, Lipton, Ouaknine, and Worrell, fixes a positive integer constant \(c\) and asks whether there exists
\[
n=\ell p^k,\qquad \ell,k\le c,\; p\ \text{prime},
\]
such that \(u_n=0\). In the algebraic-coefficient setting they also consider the adjusted index form
\[
n=\ell\,p^{k f(p)},
\]
where \(f(p)\) is the inertial degree of an unramified rational prime \(p\) in the relevant ring of integers [2006.07432].

The continuous bounded Skolem problem replaces recurrence sequences by solutions of linear differential equations. Given algebraic coefficients \(a_0,\ldots,a_{d-1}\), algebraic initial data, and a rational interval \([a,b]\), one asks whether the unique real-valued solution \(f:\mathbb R_{\ge 0}\to\mathbb R\) of
\[
f^{(d)}(t)+a_{d-1}f^{(d-1)}(t)+\cdots+a_0f(t)=0
\]
has a zero in \([a,b]\) [1506.00695].

## 2. Discrete bounded Skolem problem for linear recurrences

For fixed order, the bounded discrete problem admits a structurally explicit algorithm based on \(p\)-adic interpolation. The starting point is to choose a prime \(p>d+1\) not dividing \(a_0\), so that the companion matrix \(A\) of the recurrence lies in \(GL_d(\mathbb Z/p\mathbb Z)\) and therefore has finite order \(M\) modulo \(p\), with \(M<p^{d^2}\). The sequence is then split into \(M\) interleaved subsequences
\[
S_\ell(n)=u_{Mn+\ell},\qquad \ell=0,\ldots,M-1.
\]
Each \(S_\ell\) extends uniquely to a \(p\)-adic analytic function
\[
F_\ell:\mathbb Z_p\to\mathbb Z_p,\qquad F_\ell(n)=u_{Mn+\ell}\ \ (n\in\mathbb N),
\]
represented by a convergent Mahler series
\[
F_\ell(x)=\sum_{k=0}^\infty \beta_{\ell,k}\binom{x}{k},\qquad \beta_{\ell,k}\in\mathbb Z,\; v_p(\beta_{\ell,k}/k!)\to\infty.
\]
This interpolation reduces the bounded Skolem problem to locating \(p\)-adic zeros of finitely many analytic functions [2507.11234].

Zero counting is controlled by the \(p\)-adic Weierstrass Preparation Theorem. For a nonzero convergent power series \(F(x)=\sum_{j=0}^\infty b_jx^j\), define
\[
V(F)=\min_j v_p(b_j),\qquad j(F)=\max\{\,j:v_p(b_j)=V(F)\}.
\]
Then \(F\) has exactly \(j(F)\) zeros in the unit disc \(O_p\), counted with multiplicity. More generally, for any residue disc \(\overline D(z,r)=\{x:v_p(x-z)\ge r\}\), if \(h_{z,r}(x)=F(p^r x+z)\), then \(F\) has exactly \(j(h_{z,r})\) zeros in \(\overline D(z,r)\) [2507.11234].

A key quantitative ingredient is the van der Poorten–Schlickewei refinement of Skolem–Mahler–Lech: for each \(\ell\),
\[
j(F_\ell)\le d-1.
\]
Consequently each \(F_\ell\) has at most \(d-1\) zeros in \(O_p\), and after refinement into discs of radius \(p^{-r}\), each subsequence yields at most \(d-1\) candidate residues modulo \(p^r\). This converts a search over exponentially many integers into a search over polynomially many \(p\)-adic candidate classes when \(d\) is fixed [2507.11234].

The final verification stage does not evaluate \(u_n\) directly at huge indices. Instead one writes
\[
u_n=\alpha A^n\beta,
\]
with \(A\) the companion matrix and \(\alpha,\beta\) fixed vectors, constructs by repeated squaring an arithmetic circuit \(C_n\) of size polynomial in \(|u|+|N|\) computing \(u_n\), and combines the candidate circuits into
\[
C=C_{n_1}\times C_{n_2}\times\cdots\times C_{n_m}.
\]
Identity testing for the resulting circuit is then handled via EqSLP in \(\mathsf{coRP}\) using a Schwartz–Zippel type random evaluation test [2507.11234].

## 3. Prime-power indices and norm-based decidability

The prime-power-index specialization begins with a more general algebraic setting. Let \((u_n)_{n=0}^\infty\) be a linear recurrence sequence over the algebraic numbers of order \(d\), with characteristic roots \(\lambda_1,\ldots,\lambda_m\) that are algebraic integers in a number field \(K\), and suppose
\[
u_n=\sum_{i=1}^m A_i(n)\lambda_i^n
\]
with \(A_i(x)\in\mathcal O[x]\), where \(\mathcal O\) is the ring of integers of \(K\). The associated simple sequence is
\[
v_n=\sum_{i=1}^m A_i(0)\lambda_i^n.
\]
The bounded prime-power problem asks whether there exists \(n=\ell p^k\), with \(\ell,k\in\{0,1,\dots,c\}\) and \(p\) prime, such that \(u_n=0\) [2006.07432].

In the simple integer-coefficient case, where
\[
u_n=\sum_{i=1}^m A_i\lambda_i^n,\qquad A_i\in\mathbb Z,
\]
and the \(\lambda_i\) are distinct algebraic integers, decidability for indices \(n=p^k\) with \(k\le c\) follows from a congruence argument. The stated intuition is a “freshman’s-dream” congruence modulo \(p\), showing \(u_1^p\equiv u_p\pmod p\). Hence any prime \(p\) witnessing \(u_{p^k}=0\) must divide the norm of \(u_1\), reducing the search to finitely many primes [2006.07432].

For polynomial coefficients over \(\mathbb Z\), the method passes through the associated simple sequence \(v_n\). One first tests \(u_\ell=0\) for \(\ell\le c\). For each \(\ell\) with \(v_\ell\neq 0\), a two-step congruence argument shows that if \(u_{\ell p}=0\), then \(p\) divides the norm of \(v_\ell\), giving an explicit bound on the candidate primes. The same mechanism is iterated for higher prime powers \(p^k\) [2006.07432].

In the algebraic-coefficient case, rational primes must be adjusted by inertia. For each unramified rational prime \(p\), with inertial degree \(f(p)\), one can decide whether there exists
\[
n=\ell\,p^{k f(p)},\qquad \ell,k\le c,\; p\text{ prime},
\]
such that \(u_n=0\). The key lemma is a generalized Fermat congruence: for all \(\phi\in\mathcal O\), for unramified \(p\) and \(f=f(p)\),
\[
\phi^{p^f}-\phi\in p\mathcal O.
\]
This yields a congruence relating \(v_\ell\) and \(u_{\ell p^f}\) modulo \(p\), and thereby again restricts candidate primes to those dividing explicit norms [2006.07432].

The resulting decision procedure computes the associated simple sequence, forms
\[
L_{(c)}=\{\ell:0\le \ell\le c\ \text{and}\ v_\ell\neq 0\},
\]
computes the algebraic-integer norms \(N_\ell=N_K(v_\ell)\), factors each \(N_\ell\), and checks only those primes dividing \(N_\ell\). For each such \(p\) and each \(k\le c\), one evaluates \(u_n\) at
\[
n=\ell p^k\quad \text{or}\quad n=\ell p^{k f(p)},
\]
for example by fast doubling or matrix exponentiation in \(O(\log n)\) arithmetic operations. Correctness rests on the congruences showing that any prime capable of producing a zero must divide one of the norms \(N_\ell\) [2006.07432].

## 4. Complexity landscape and lower bounds

The complexity picture differs sharply across bounded Skolem variants. For the standard bounded discrete problem, the 2025 result gives a randomised polynomial-time algorithm for every fixed order \(d\). More precisely, the depth-\(R\) search over \(p\)-adic discs uses
\[
R=\lceil \log_p(N+1)\rceil,
\]
and each subsequence contributes at most \(d-1\) surviving discs per level. Since \(M,p=\mathrm{poly}(|u|)\), the total number of integer candidates is polynomial in \(|u|\), and the overall running time is polynomial in \(|u|+|N|\) for fixed \(d\). The dependence on \(d\), however, is exponential; the same work states that this appears necessary because the bounded Skolem problem is \(\mathsf{NP}\)-hard when the order is unbounded [2507.11234].

The same paper derives a low-order corollary for the unrestricted Skolem Problem. Because the unrestricted problem for order at most \(4\) reduces in polynomial time to the bounded version, and bounded Skolem at order at most \(4\) lies in \(\mathsf{coRP}\), it follows that the unrestricted Skolem Problem for LRS of order at most \(4\) lies in \(\mathsf{coRP}\), improving the previous upper bound of \(\mathsf{NP}^{\mathsf{RP}}\) [2507.11234].

The prime-power-index problem has a different complexity profile. If \(S\) denotes the bit-size of the input, then the norm \(N_\ell\) to be factored satisfies
\[
N_\ell \le (\mathrm{poly}(S))^{\exp(S)},
\]
its prime factorization can be done in time subexponential in \(N_\ell\), and the total number of candidate primes is bounded by \(O(\exp(S))\). The stated overall runtime bound is therefore
\[
2^{2^{O(S)}}.
\]
Thus the bounded Skolem problem for prime-power indices is decidable in doubly-exponential time [2006.07432].

Decidability does not imply tractability. Kenison–Lipton–Ouaknine–Worrell show \(\mathsf{NP}\)-hardness even for cyclotomic sequences, that is, sequences in which all \(\lambda_i\) are roots of unity and \(A_i\in\mathbb Z\). In that restricted setting, deciding whether there exists a prime \(p\) with \(u_p=0\) is already \(\mathsf{NP}\)-hard via a reduction from Subset-Sum [2006.07432].

A recurring misconception is that a bounded search domain should make the problem routine. The available results do not support that conclusion. In the fixed-order discrete case, the challenge is to detect zeros in an exponentially large interval while keeping the number of candidates polynomial. In the prime-power case, even after reducing the prime search to divisors of explicitly constructed norms, the worst-case decision procedure remains doubly exponential [2507.11234] [2006.07432].

## 5. Continuous bounded Skolem problem

The continuous version concerns zeros of solutions to linear differential equations rather than recurrences. If
\[
f^{(d)}(t)+a_{d-1}f^{(d-1)}(t)+\cdots+a_0f(t)=0
\]
has characteristic roots \(\lambda_1,\ldots,\lambda_m\), then the unique solution can be written as
\[
f(t)=\sum_{j=1}^m P_j(t)e^{\lambda_j t},
\]
where the degree of \(P_j\) is the multiplicity of \(\lambda_j\) minus one. Writing \(\lambda_j=r_j+i\omega_j\), one often rewrites the solution as
\[
f(t)=\sum_{j=1}^m e^{r_j t}\bigl(Q_{1,j}(t)\sin(\omega_j t)+Q_{2,j}(t)\cos(\omega_j t)\bigr),
\]
and calls the \(\omega_j\) the frequencies of \(f\) [1506.00695].

The principal decidability theorem is conditional. Assuming Schanuel’s Conjecture, there is an algorithm that, given a rational interval \([a,b]\), a linear differential equation with real algebraic coefficients, and algebraic initial data, decides whether the solution has a zero in \([a,b]\). The proof factors an associated Laurent polynomial in the ring
\[
\mathcal R=K[x,y_1,y_1^{-1},\ldots,y_r,y_r^{-1},z_1,z_1^{-1},\ldots,z_s,z_s^{-1}],
\]
with \(y_k=e^{a_k t}\) and \(z_\ell=e^{i b_\ell t}\), and then classifies irreducible factors by conjugation into three types [1506.00695].

For Type-1 factors, \(P\) and \(\overline P\) are coprime. Under Schanuel’s Conjecture, simultaneous vanishing of the corresponding equations is excluded, so no zeros occur. For Type-2 factors, where \(P=\overline P\), the function is real-valued; one shows that \(f'\) is represented by another Laurent polynomial coprime with \(P\), whence \(f\) has only simple zeros, enabling a standard zero-finding procedure based on Lipschitz continuity and effective approximation. For Type-3 factors, where \(P=Q+z^u\overline Q\), one writes \(f(t)=g_1(t)+g_2(t)\) with \(g_2(t)=z^u\overline{g_1(t)}\), defines
\[
h(t)=\pi+i\,\mathrm{Log}\!\bigl(g_1(t)/g_2(t)\bigr),
\]
and reduces zero existence for \(f\) to zero existence for \(h\), again using that tangential zeros are absent and \(|h|\) is Lipschitz away from the branch cut [1506.00695].

The bounded continuous problem remains markedly less explicit than the fixed-order discrete case. The procedure is effective in the classical Turing model provided one assumes Schanuel’s Conjecture as an oracle for transcendence-degree judgments, but no elementary bound on the required numeric precision is obtained, and no explicit time-complexity bound is known. Accordingly, the problem is not placed in any known complexity class unconditionally [1506.00695].

The same work relates bounded and unbounded continuous Skolem problems via frequency structure. The unbounded problem can be reduced to the bounded problem if there is at most one rationally linearly independent frequency, or if there are two rationally linearly independent frequencies and all characteristic roots are simple. By contrast, decidability of the unbounded problem in the case of two or more rationally linearly independent frequencies would imply computability of the Diophantine-approximation types of all real algebraic numbers [1506.00695].

## 6. Open directions, limitations, and terminological distinctions

Several open problems remain explicit in the current literature. For the prime-power-index problem, one stated objective is to close the gap between the exponential lower bound on the smallest possible witnessing prime and the doubly-exponential upper bound produced by the norm-factoring algorithm. Other stated directions are extension beyond prime-power indices to patterns such as
\[
n=\ell p^k q^m
\]
and improvement of the complexity upper bound, for example to single-exponential or PSPACE, or else proving stronger lower bounds [2006.07432].

For the standard bounded discrete problem, the low-order complexity paper asks whether there is an effective singly-exponential upper bound on the largest zero of a non-degenerate LRS of arbitrary fixed order. The paper notes that such a bound would collapse the general Skolem Problem to the bounded version and yield \(\mathsf{coRP}\) membership at that order. It also leaves open the full decidability of the unrestricted Skolem Problem for orders at least \(5\) [2507.11234].

In the continuous setting, the central limitation is conditionality: bounded continuous Skolem is decidable only under Schanuel’s Conjecture in the cited work, and the absence of an elementary precision bound prevents a finer unconditional complexity classification. This suggests that transcendence-theoretic obstructions are intrinsic to the present methods rather than incidental [1506.00695].

A separate terminological issue arises from logic and database theory. There, “Skolem” may refer to the semi-oblivious chase, also called the Skolem chase, and one studies whether a ruleset is \(k\)-bounded in chase depth. That problem concerns existential rules, breadth-first derivation rank, and universal models; it is unrelated to zero detection in linear recurrences or differential equations. The overlap is only lexical, not conceptual [2004.10030].

Taken together, these results show that “bounded Skolem problem” names a technically coherent but methodologically diverse cluster of zero-detection problems. In the discrete finite-order setting, \(p\)-adic analytic structure yields randomised polynomial time for fixed order. In the prime-power specialization, congruence arguments and algebraic norms give decidability in doubly-exponential time. In the continuous setting, bounded-domain decidability is presently conditional on Schanuel’s Conjecture. The differences among these outcomes reflect genuine structural differences in the arithmetic and analytic objects under study rather than merely differences in presentation [2507.11234] [2006.07432] [1506.00695].

Source: https://www.emergentmind.com/topics/bounded-skolem-problem