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A deterministic polynomial-time algorithm outputs a list of constant-depth circuits containing all factors of an n-variate s-sparse polynomial with bounded individual degree d, possibly with spurious extras.

2026-06-26 01:42 UTC pith:AB5RKMH3

load-bearing objection The paper gives a deterministic poly(n, s^d) algorithm outputting constant-depth circuits that cover all factors of bounded-ind-degree sparse polynomials, plus a quasipoly algorithm and new factor-count bound for the general case.

arxiv 2606.27293 v1 pith:AB5RKMH3 submitted 2026-06-25 cs.CC

Deterministic Algorithms for Low Individual Degree Factors of Sparse Polynomials

Somnath Bhattacharjee , Rishabh Kothary , Shanthanu S. Rai , Shubhangi Saraf This is my paper
classification cs.CC
keywords sparse polynomial factorizationdeterministic algorithmsindividual degreealgebraic circuitsmultivariate polynomialscomputational complexityconstant depth circuits
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops deterministic algorithms for factoring sparse multivariate polynomials when individual degrees are bounded. The core procedure runs in time polynomial in n and s to the power d, producing constant-depth circuits that are guaranteed to include every factor though the list may contain extras. It applies over fields of characteristic zero or large enough characteristic and generalizes earlier deterministic recovery of only sparse factors. A second result gives a quasipolynomial-time procedure that finds bounded individual degree factors even when the input polynomial has unbounded individual degree. These advances matter because they supply explicit deterministic procedures in a setting where randomized methods have long been standard.

Core claim

We give a deterministic polynomial-time algorithm which takes as input an n-variate s-sparse polynomial f of bounded individual degree d and outputs a list of circuits which contains all factors of f, although there might be additional spurious circuits in the list. The algorithm runs in time poly(n, s^d). Additionally, every circuit in the list has constant depth. Our algorithm works over all fields of characteristic 0 or sufficiently large characteristic. As a corollary, all factors can be recovered in poly(n, s^{d^2 log n}) time, and the result recovers earlier work on sparse factors.

What carries the argument

The deterministic procedure that constructs a list of constant-depth circuits guaranteed to include all true factors of the input sparse polynomial.

Load-bearing premise

The underlying field has characteristic zero or is sufficiently large.

What would settle it

An explicit n-variate s-sparse polynomial over a large-characteristic field together with a factor of individual degree at most d that the algorithm either misses or fails to output within poly(n, s^d) time.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • All factors of f can be recovered in time poly(n, s^{d^2 log n}) by combining the list with known interpolation and divisibility testing.
  • The earlier algorithmic results of Bhargava, Saraf and Volkovich and the improvement by Chuyoon and Shpilka are recovered as special cases.
  • A new upper bound holds on the total number of bounded individual degree factors of any sparse polynomial.
  • Previous quasipolynomial algorithms for recovering bounded total degree factors are strengthened to the individual degree setting.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The constant-depth guarantee on the output circuits may allow direct use inside constant-depth circuit classes for related algebraic tasks.
  • The new counting bound on bounded individual degree factors could be applied to bound the size of factor sets in other sparse algebraic objects.
  • Extending the polynomial-time guarantee to small-characteristic fields would require new techniques that avoid the current dependence on field size.
  • The separation between bounded individual degree and bounded total degree may be useful for studying factorizations where total degree is large but individual degrees remain small.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 3 minor

Summary. The manuscript claims two deterministic algorithms for factoring sparse polynomials. Result 1 gives a poly(n, s^d)-time algorithm, over fields of char 0 or sufficiently large char, that on input an n-variate s-sparse f of individual degree d outputs a list of constant-depth circuits containing every factor of f (possibly with extra spurious circuits). Result 2 gives a quasipolynomial-time algorithm, over arbitrary fields, that on a general s-sparse f of individual degree D outputs all factors of individual degree at most d (again possibly with spurious entries), running in time poly(D^{d log s}, s^{d^2 log n}). Both results are obtained by generalizing the Chuyoon-Shpilka sparse-factor algorithm via interpolation and divisibility testing; corollaries recover all factors in poly(n, s^{d^2 log n}) time and strengthen prior work of Dutta-Sinhababu-Thierauf and Kumar-Ramanathan-Saptharishi.

Significance. If the proofs hold, the results advance deterministic algebraic algorithms for sparse polynomials by relaxing the sparsity requirement on the output factors while preserving polynomial or quasipolynomial time; the explicit handling of spurious circuits and the new upper bound on the number of bounded-individual-degree factors are useful additions. The work correctly cites and builds on the interpolation/divisibility toolkit without introducing hidden parameters or circular reductions.

minor comments (3)
  1. [§1] §1, paragraph after Theorem 1.2: the statement that the list 'contains all factors' should explicitly cross-reference the precise definition of 'factor' used in the divisibility-testing subroutine (e.g., whether monic or up to units).
  2. The running-time expression in Result 2 contains both D^{d log s} and s^{d^2 log n}; a short remark clarifying which term dominates when d is treated as a fixed constant versus when d grows would improve readability.
  3. The bibliography entry for Chuyoon-Shpilka should include the full conference or journal details and year to match the citation style used for Bhargava-Saraf-Volkovich.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment, the accurate summary of our results, and the recommendation for minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central result is a new deterministic poly(n, s^d)-time algorithm for listing (possibly with spurious) constant-depth circuits containing all factors of an n-variate s-sparse input of individual degree d. This is obtained by generalizing the Chuyoon-Shpilka sparse-factor algorithm via standard interpolation and divisibility-testing techniques, with the characteristic hypothesis stated explicitly. The self-citation to Bhargava-Saraf-Volkovich is used only to recover their prior result as a corollary of the new algorithm; the new claim does not reduce to or depend on that citation for its correctness. No self-definitional steps, fitted-input predictions, ansatz smuggling, or uniqueness theorems imported from the same authors appear in the derivation chain. The algorithm is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard algebraic axioms (polynomial rings over fields, unique factorization) and prior algorithmic primitives (interpolation, divisibility testing) mentioned in the abstract; no free parameters or invented entities are introduced.

axioms (2)
  • standard math Polynomials over a field form a unique factorization domain
    Factoring algorithms rely on this algebraic property.
  • domain assumption Efficient interpolation and divisibility testing are available for the relevant circuit classes
    The corollaries explicitly invoke these known techniques to recover all factors from the output list.

pith-pipeline@v0.9.1-grok · 5930 in / 1417 out tokens · 39696 ms · 2026-06-26T01:42:03.462342+00:00 · methodology

0 comments
read the original abstract

We study factoring algorithms for general sparse polynomials and sparse polynomials of bounded individual degree and prove the following results. 1. We give a deterministic polynomial-time algorithm which takes as input an $n$-variate $s$-sparse polynomial $f$ of bounded individual degree $d$ and outputs a list of circuits which contains all factors of $f$, although there might be additional spurious circuits in the list. The algorithm runs in time $\operatorname{poly}(n, s^d)$. Additionally, every circuit in the list has constant depth. Our algorithm works over all fields of characteristic 0 or sufficiently large characteristic. Our result generalizes a recent result of Chuyoon and Shpilka that gives a $\operatorname{poly}(n, s^d)$-time algorithm for recovering all sparse factors of $f$ (without spurious factors). As a corollary, we can also recover all factors of $f$ in time $\operatorname{poly}(n, s^{d^2 \log n})$, and recover the algorithmic result of Bhargava, Saraf and Volkovich and its improvement by Chuyoon and Shpilka. Both the above consequences follow from known interpolation and divisibility testing techniques. 2. We give a deterministic quasipolynomial-time algorithm which takes as input a general $n$-variate $s$-sparse polynomial $f$ of (unbounded) individual degree $D$ and outputs a list of polynomials which contains all factors of $f$ that have bounded individual degree $d$. The algorithm runs in time $\operatorname{poly}(D^{d \log s}, s^{d^2 \log n})$ and works over arbitrary fields. The list may again contain spurious elements. Our result strengthens results of Dutta, Sinhababu and Thierauf and Kumar, Ramanathan and Saptharishi which give algorithms to recover all factors of $f$ of bounded total degree. A consequence of our algorithm is a new upper bound on the total number of bounded individual degree factors of a sparse polynomial.

discussion (0)

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