Bounded Skolem Problem: Complexity & Algorithms

• The bounded Skolem problem is a decision problem family that determines whether linear recurrences or differential equation solutions reach zero within a prescribed bounded domain.
• It utilizes advanced techniques such as p-adic analytic methods and congruence arguments to reduce an exponentially large search space to a manageable set when the order is fixed.
• The problem exhibits diverse computational complexity, ranging from randomized polynomial time for fixed-order discrete cases to doubly exponential time for prime-power indices and conditionally decidable continuous variants.

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 $[a,b]$ (Bacik et al., 15 Jul 2025, Kenison et al., 2020, Chonev et al., 2015).

1. Formal setting and principal variants

The classical Skolem Problem asks, for a linear recurrence sequence $(u_n)$, whether there exists any $\langle u_n\rangle_{n=0}^\infty$0 such that $\langle u_n\rangle_{n=0}^\infty$1. For an integer LRS of order $\langle u_n\rangle_{n=0}^\infty$2, the recurrence has the form

$\langle u_n\rangle_{n=0}^\infty$3

with integer coefficients $\langle u_n\rangle_{n=0}^\infty$4, and the minimal such $\langle u_n\rangle_{n=0}^\infty$5 is called the order of the LRS. If the characteristic polynomial

$\langle u_n\rangle_{n=0}^\infty$6

has distinct algebraic roots $\langle u_n\rangle_{n=0}^\infty$7 with multiplicities $\langle u_n\rangle_{n=0}^\infty$8, then the closed form is

$\langle u_n\rangle_{n=0}^\infty$9

where each $N\in\mathbb N$0 has degree $N\in\mathbb N$1 (Bacik et al., 15 Jul 2025).

The bounded discrete problem restricts the search space to a finite initial segment. Its computational content is nontrivial because the bound $N\in\mathbb N$2 is part of the input in binary, so the interval $N\in\mathbb N$3 may be exponentially large in the input length. The 2025 low-order complexity work explicitly studies this version and shows that, for every fixed $N\in\mathbb N$4, there is a randomised polynomial-time algorithm on LRS of order at most $N\in\mathbb N$5 (Bacik et al., 15 Jul 2025).

A different bounded variant, studied by Kenison, Lipton, Ouaknine, and Worrell, fixes a positive integer constant $N\in\mathbb N$6 and asks whether there exists

$N\in\mathbb N$7

such that $N\in\mathbb N$8. In the algebraic-coefficient setting they also consider the adjusted index form

$N\in\mathbb N$9

where $n\in\{0,\ldots,N\}$0 is the inertial degree of an unramified rational prime $n\in\{0,\ldots,N\}$1 in the relevant ring of integers (Kenison et al., 2020).

The continuous bounded Skolem problem replaces recurrence sequences by solutions of linear differential equations. Given algebraic coefficients $n\in\{0,\ldots,N\}$2, algebraic initial data, and a rational interval $n\in\{0,\ldots,N\}$3, one asks whether the unique real-valued solution $n\in\{0,\ldots,N\}$4 of

$n\in\{0,\ldots,N\}$5

has a zero in $n\in\{0,\ldots,N\}$6 (Chonev et al., 2015).

2. Discrete bounded Skolem problem for linear recurrences

For fixed order, the bounded discrete problem admits a structurally explicit algorithm based on $n\in\{0,\ldots,N\}$7-adic interpolation. The starting point is to choose a prime $n\in\{0,\ldots,N\}$8 not dividing $n\in\{0,\ldots,N\}$9, so that the companion matrix $u_n=0$0 of the recurrence lies in $u_n=0$1 and therefore has finite order $u_n=0$2 modulo $u_n=0$3, with $u_n=0$4. The sequence is then split into $u_n=0$5 interleaved subsequences

$u_n=0$6

Each $u_n=0$7 extends uniquely to a $u_n=0$8-adic analytic function

$u_n=0$9

represented by a convergent Mahler series

$n=\ell p^k$0

This interpolation reduces the bounded Skolem problem to locating $n=\ell p^k$1-adic zeros of finitely many analytic functions (Bacik et al., 15 Jul 2025).

Zero counting is controlled by the $n=\ell p^k$2-adic Weierstrass Preparation Theorem. For a nonzero convergent power series $n=\ell p^k$3, define

$n=\ell p^k$4

Then $n=\ell p^k$5 has exactly $n=\ell p^k$6 zeros in the unit disc $n=\ell p^k$7, counted with multiplicity. More generally, for any residue disc $n=\ell p^k$8, if $n=\ell p^k$9, then $\ell,k\le c$0 has exactly $\ell,k\le c$1 zeros in $\ell,k\le c$2 (Bacik et al., 15 Jul 2025).

A key quantitative ingredient is the van der Poorten–Schlickewei refinement of Skolem–Mahler–Lech: for each $\ell,k\le c$3,

$\ell,k\le c$4

Consequently each $\ell,k\le c$5 has at most $\ell,k\le c$6 zeros in $\ell,k\le c$7, and after refinement into discs of radius $\ell,k\le c$8, each subsequence yields at most $\ell,k\le c$9 candidate residues modulo $p$0. This converts a search over exponentially many integers into a search over polynomially many $p$1-adic candidate classes when $p$2 is fixed (Bacik et al., 15 Jul 2025).

The final verification stage does not evaluate $p$3 directly at huge indices. Instead one writes

$p$4

with $p$5 the companion matrix and $p$6 fixed vectors, constructs by repeated squaring an arithmetic circuit $p$7 of size polynomial in $p$8 computing $p$9, and combines the candidate circuits into

$[a,b]$0

Identity testing for the resulting circuit is then handled via EqSLP in $[a,b]$1 using a Schwartz–Zippel type random evaluation test (Bacik et al., 15 Jul 2025).

3. Prime-power indices and norm-based decidability

The prime-power-index specialization begins with a more general algebraic setting. Let $[a,b]$2 be a linear recurrence sequence over the algebraic numbers of order $[a,b]$3, with characteristic roots $[a,b]$4 that are algebraic integers in a number field $[a,b]$5, and suppose

$[a,b]$6

with $[a,b]$7, where $[a,b]$8 is the ring of integers of $[a,b]$9. The associated simple sequence is

$(u_n)$0

The bounded prime-power problem asks whether there exists $(u_n)$1, with $(u_n)$2 and $(u_n)$3 prime, such that $(u_n)$4 (Kenison et al., 2020).

In the simple integer-coefficient case, where

$(u_n)$5

and the $(u_n)$6 are distinct algebraic integers, decidability for indices $(u_n)$7 with $(u_n)$8 follows from a congruence argument. The stated intuition is a “freshman’s-dream” congruence modulo $(u_n)$9, showing $\langle u_n\rangle_{n=0}^\infty$00. Hence any prime $\langle u_n\rangle_{n=0}^\infty$01 witnessing $\langle u_n\rangle_{n=0}^\infty$02 must divide the norm of $\langle u_n\rangle_{n=0}^\infty$03, reducing the search to finitely many primes (Kenison et al., 2020).

For polynomial coefficients over $\langle u_n\rangle_{n=0}^\infty$04, the method passes through the associated simple sequence $\langle u_n\rangle_{n=0}^\infty$05. One first tests $\langle u_n\rangle_{n=0}^\infty$06 for $\langle u_n\rangle_{n=0}^\infty$07. For each $\langle u_n\rangle_{n=0}^\infty$08 with $\langle u_n\rangle_{n=0}^\infty$09, a two-step congruence argument shows that if $\langle u_n\rangle_{n=0}^\infty$10, then $\langle u_n\rangle_{n=0}^\infty$11 divides the norm of $\langle u_n\rangle_{n=0}^\infty$12, giving an explicit bound on the candidate primes. The same mechanism is iterated for higher prime powers $\langle u_n\rangle_{n=0}^\infty$13 (Kenison et al., 2020).

In the algebraic-coefficient case, rational primes must be adjusted by inertia. For each unramified rational prime $\langle u_n\rangle_{n=0}^\infty$14, with inertial degree $\langle u_n\rangle_{n=0}^\infty$15, one can decide whether there exists

$\langle u_n\rangle_{n=0}^\infty$16

such that $\langle u_n\rangle_{n=0}^\infty$17. The key lemma is a generalized Fermat congruence: for all $\langle u_n\rangle_{n=0}^\infty$18, for unramified $\langle u_n\rangle_{n=0}^\infty$19 and $\langle u_n\rangle_{n=0}^\infty$20,

$\langle u_n\rangle_{n=0}^\infty$21

This yields a congruence relating $\langle u_n\rangle_{n=0}^\infty$22 and $\langle u_n\rangle_{n=0}^\infty$23 modulo $\langle u_n\rangle_{n=0}^\infty$24, and thereby again restricts candidate primes to those dividing explicit norms (Kenison et al., 2020).

The resulting decision procedure computes the associated simple sequence, forms

$\langle u_n\rangle_{n=0}^\infty$25

computes the algebraic-integer norms $\langle u_n\rangle_{n=0}^\infty$26, factors each $\langle u_n\rangle_{n=0}^\infty$27, and checks only those primes dividing $\langle u_n\rangle_{n=0}^\infty$28. For each such $\langle u_n\rangle_{n=0}^\infty$29 and each $\langle u_n\rangle_{n=0}^\infty$30, one evaluates $\langle u_n\rangle_{n=0}^\infty$31 at

$\langle u_n\rangle_{n=0}^\infty$32

for example by fast doubling or matrix exponentiation in $\langle u_n\rangle_{n=0}^\infty$33 arithmetic operations. Correctness rests on the congruences showing that any prime capable of producing a zero must divide one of the norms $\langle u_n\rangle_{n=0}^\infty$34 (Kenison et al., 2020).

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 $\langle u_n\rangle_{n=0}^\infty$35. More precisely, the depth-$\langle u_n\rangle_{n=0}^\infty$36 search over $\langle u_n\rangle_{n=0}^\infty$37-adic discs uses

$\langle u_n\rangle_{n=0}^\infty$38

and each subsequence contributes at most $\langle u_n\rangle_{n=0}^\infty$39 surviving discs per level. Since $\langle u_n\rangle_{n=0}^\infty$40, the total number of integer candidates is polynomial in $\langle u_n\rangle_{n=0}^\infty$41, and the overall running time is polynomial in $\langle u_n\rangle_{n=0}^\infty$42 for fixed $\langle u_n\rangle_{n=0}^\infty$43. The dependence on $\langle u_n\rangle_{n=0}^\infty$44, however, is exponential; the same work states that this appears necessary because the bounded Skolem problem is $\langle u_n\rangle_{n=0}^\infty$45-hard when the order is unbounded (Bacik et al., 15 Jul 2025).

The same paper derives a low-order corollary for the unrestricted Skolem Problem. Because the unrestricted problem for order at most $\langle u_n\rangle_{n=0}^\infty$46 reduces in polynomial time to the bounded version, and bounded Skolem at order at most $\langle u_n\rangle_{n=0}^\infty$47 lies in $\langle u_n\rangle_{n=0}^\infty$48, it follows that the unrestricted Skolem Problem for LRS of order at most $\langle u_n\rangle_{n=0}^\infty$49 lies in $\langle u_n\rangle_{n=0}^\infty$50, improving the previous upper bound of $\langle u_n\rangle_{n=0}^\infty$51 (Bacik et al., 15 Jul 2025).

The prime-power-index problem has a different complexity profile. If $\langle u_n\rangle_{n=0}^\infty$52 denotes the bit-size of the input, then the norm $\langle u_n\rangle_{n=0}^\infty$53 to be factored satisfies

$\langle u_n\rangle_{n=0}^\infty$54

its prime factorization can be done in time subexponential in $\langle u_n\rangle_{n=0}^\infty$55, and the total number of candidate primes is bounded by $\langle u_n\rangle_{n=0}^\infty$56. The stated overall runtime bound is therefore

$\langle u_n\rangle_{n=0}^\infty$57

Thus the bounded Skolem problem for prime-power indices is decidable in doubly-exponential time (Kenison et al., 2020).

Decidability does not imply tractability. Kenison–Lipton–Ouaknine–Worrell show $\langle u_n\rangle_{n=0}^\infty$58-hardness even for cyclotomic sequences, that is, sequences in which all $\langle u_n\rangle_{n=0}^\infty$59 are roots of unity and $\langle u_n\rangle_{n=0}^\infty$60. In that restricted setting, deciding whether there exists a prime $\langle u_n\rangle_{n=0}^\infty$61 with $\langle u_n\rangle_{n=0}^\infty$62 is already $\langle u_n\rangle_{n=0}^\infty$63-hard via a reduction from Subset-Sum (Kenison et al., 2020).

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 (Bacik et al., 15 Jul 2025, Kenison et al., 2020).

5. Continuous bounded Skolem problem

The continuous version concerns zeros of solutions to linear differential equations rather than recurrences. If

$\langle u_n\rangle_{n=0}^\infty$64

has characteristic roots $\langle u_n\rangle_{n=0}^\infty$65, then the unique solution can be written as

$\langle u_n\rangle_{n=0}^\infty$66

where the degree of $\langle u_n\rangle_{n=0}^\infty$67 is the multiplicity of $\langle u_n\rangle_{n=0}^\infty$68 minus one. Writing $\langle u_n\rangle_{n=0}^\infty$69, one often rewrites the solution as

$\langle u_n\rangle_{n=0}^\infty$70

and calls the $\langle u_n\rangle_{n=0}^\infty$71 the frequencies of $\langle u_n\rangle_{n=0}^\infty$72 (Chonev et al., 2015).

The principal decidability theorem is conditional. Assuming Schanuel’s Conjecture, there is an algorithm that, given a rational interval $\langle u_n\rangle_{n=0}^\infty$73, a linear differential equation with real algebraic coefficients, and algebraic initial data, decides whether the solution has a zero in $\langle u_n\rangle_{n=0}^\infty$74. The proof factors an associated Laurent polynomial in the ring

$\langle u_n\rangle_{n=0}^\infty$75

with $\langle u_n\rangle_{n=0}^\infty$76 and $\langle u_n\rangle_{n=0}^\infty$77, and then classifies irreducible factors by conjugation into three types (Chonev et al., 2015).

For Type-1 factors, $\langle u_n\rangle_{n=0}^\infty$78 and $\langle u_n\rangle_{n=0}^\infty$79 are coprime. Under Schanuel’s Conjecture, simultaneous vanishing of the corresponding equations is excluded, so no zeros occur. For Type-2 factors, where $\langle u_n\rangle_{n=0}^\infty$80, the function is real-valued; one shows that $\langle u_n\rangle_{n=0}^\infty$81 is represented by another Laurent polynomial coprime with $\langle u_n\rangle_{n=0}^\infty$82, whence $\langle u_n\rangle_{n=0}^\infty$83 has only simple zeros, enabling a standard zero-finding procedure based on Lipschitz continuity and effective approximation. For Type-3 factors, where $\langle u_n\rangle_{n=0}^\infty$84, one writes $\langle u_n\rangle_{n=0}^\infty$85 with $\langle u_n\rangle_{n=0}^\infty$86, defines

$\langle u_n\rangle_{n=0}^\infty$87

and reduces zero existence for $\langle u_n\rangle_{n=0}^\infty$88 to zero existence for $\langle u_n\rangle_{n=0}^\infty$89, again using that tangential zeros are absent and $\langle u_n\rangle_{n=0}^\infty$90 is Lipschitz away from the branch cut (Chonev et al., 2015).

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 (Chonev et al., 2015).

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 (Chonev et al., 2015).

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

$\langle u_n\rangle_{n=0}^\infty$91

and improvement of the complexity upper bound, for example to single-exponential or PSPACE, or else proving stronger lower bounds (Kenison et al., 2020).

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 $\langle u_n\rangle_{n=0}^\infty$92 membership at that order. It also leaves open the full decidability of the unrestricted Skolem Problem for orders at least $\langle u_n\rangle_{n=0}^\infty$93 (Bacik et al., 15 Jul 2025).

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 (Chonev et al., 2015).

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 $\langle u_n\rangle_{n=0}^\infty$94-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 (Delivorias et al., 2020).

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, $\langle u_n\rangle_{n=0}^\infty$95-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 (Bacik et al., 15 Jul 2025, Kenison et al., 2020, Chonev et al., 2015).