{"id":"f631dced-32ca-4cc0-b4d5-f0dc4e12184f","arxiv_id":"2605.22514","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Probabilistic algorithm computes isolated regular solutions of composable polynomial systems with arithmetic complexity polynomial in input size and solution count.","lead":"The paper describes a probabilistic symbolic homotopy algorithm that reduces composable polynomial systems to a smaller system in auxiliary variables before lifting solutions. A smart generalist might read it for potential efficiency gains in solving structured algebraic equations arising in symmetry and invariant computations.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"Reader correctly flags provisional status from abstract-only access. Without the detailed construction, no load-bearing technical concern can be located or tested; verdict remains UNVERDICTED pending full text.","tokens_in":1757,"tokens_out":159,"duration_ms":24312,"concrete_test":"Obtain full preprint and inspect the main theorem plus complexity proof to confirm the homotopy reduction yields arithmetic complexity polynomial in input size and number of solutions.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Full text unavailable; only abstract provided. No algorithmic description, proof, or complexity analysis is accessible, so no internal assumption or step in the claimed reduction to the g_j system can be examined for failure modes.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript claims to present a probabilistic symbolic homotopy algorithm for computing all isolated regular solutions of a system of n polynomial equations in n variables over a field of characteristic zero, where the system has a composable structure (each f_i = h_i(g_1, ..., g_n)). Exploiting this structure reduces the problem to a system in the g_j variables. The algorithm is asserted to have arithmetic complexity polynomial in the input size and the number of solutions, with applications to subrings generated by algebraically independent polynomials and to invariant polynomials under finite reflection groups (e.g., S_n, B_n, exceptional groups E_6, E_7, E_8).","tokens_in":1773,"tokens_out":350,"duration_ms":60015,"significance":"If the reduction, algorithm, and complexity bound hold, the result would be significant for symbolic computation and algebraic geometry. It targets a structured class of systems that arise naturally in invariant theory, where general-purpose solvers face high complexity; a polynomial bound in input size and solution count could make previously intractable problems practical, especially for reflection-group invariants.","major_comments":[{"comment":"Abstract: the central claim that the probabilistic algorithm achieves arithmetic complexity polynomial in the input size and number of solutions is asserted without any derivation, supporting lemmas, analysis of regularity conditions, or description of the homotopy construction and reduction step; these details are required to verify the claim and are absent from the provided manuscript (which consists only of the abstract).","section":null}],"minor_comments":[],"recommendation":"uncertain","confidential_remarks":"The full manuscript text was unavailable; only the abstract was supplied for review. This prevents any substantive technical evaluation of the algorithm or proofs."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their summary and for highlighting the potential significance of the work for symbolic computation and invariant theory. We agree that the current manuscript version is limited to the abstract and therefore lacks the requested derivations and details.","responses":[{"response":"We acknowledge that the provided manuscript consists solely of the abstract and does not contain the supporting derivations, lemmas, regularity analysis, or explicit description of the homotopy construction and reduction. In the revised full manuscript we will include: (i) the precise reduction from the composable system (f_1,…,f_n) to a system in the g_j variables, (ii) the construction of the probabilistic symbolic homotopy, (iii) the regularity conditions ensuring the isolated regular solutions are captured, and (iv) the arithmetic complexity analysis establishing the polynomial bound in input size and number of solutions. These additions will allow direct verification of the central claim.","revision_made":"yes","referee_comment":"Abstract: the central claim that the probabilistic algorithm achieves arithmetic complexity polynomial in the input size and number of solutions is asserted without any derivation, supporting lemmas, analysis of regularity conditions, or description of the homotopy construction and reduction step; these details are required to verify the claim and are absent from the provided manuscript (which consists only of the abstract)."}],"tokens_in":1361,"tokens_out":287,"duration_ms":36621,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main point is that this abstract describes a probabilistic algorithm for isolated regular solutions of composable polynomial systems: each f_i is written as a composition h_i of the g_j's, the system is reduced in the g variables, and then symbolic homotopy is applied, with arithmetic complexity stated to be polynomial in input size and number of solutions. The reduction step that uses the explicit composable form looks like the actual algorithmic novelty relative to plain symbolic homotopy methods. The paper also does a clear job linking the idea to concrete settings, especially invariant rings of finite reflection groups via the Chevalley-Shephard-Todd theorem, with examples such as S_n, B_n, dihedral groups, and the exceptional groups E6 through H4. That connection is useful because those systems really do arise in practice and have the required algebraic independence of the g's. The reduction therefore targets a recurring structure rather than a generic dense system. The soft spots are straightforward and stem directly from the fact that only the abstract is available. The polynomial complexity bound is asserted but no derivation, lemmas on solution preservation, or handling of regularity conditions appear, so it is impossible to see whether the reduction step introduces hidden costs or fails on certain inputs. The central assumption that the system is composable is taken as given, which is fine for the intended class but leaves open how often real problems satisfy it exactly. No circularity or invented quantities are visible in what is shown. This paper is for researchers in symbolic computation and algebraic geometry who routinely solve structured polynomial systems, particularly those coming from invariant theory or symmetry. A reader who already works with homotopy continuation or needs speed-ups on reflection-group invariants would find the targeted reduction worth examining if the details hold. It deserves a serious referee because the idea addresses a practical bottleneck with a plausible structural shortcut and cites the right background theorems. I recommend sending it to peer review so the full algorithm, proofs, and complexity analysis can be evaluated by experts who can test the claims on examples.","headline":"The abstract sketches a reduction to exploit composable structure before symbolic homotopy, claiming polynomial complexity, but without the full paper the claim cannot be checked.","tokens_in":2229,"tokens_out":473,"would_cite":false,"duration_ms":51091,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[],"headline":"Polynomial homotopy solver for composable systems; no RS overlap","alignment":"orthogonal","rationale":"Paper's core is a probabilistic symbolic homotopy + Newton-Hensel lifting algorithm that decomposes f = h ∘ g into outer system h(Y)=0 solved via geometric resolutions, then inner lifting through g via parametric GLS_Lifting, yielding complexity polynomial in deg(h), deg(g) and SLP sizes rather than deg(f). This is standard computer-algebra machinery (RURs, straight-line programs, Bézout bounds on separate factors C and D). RS framework (reality_from_one_distinction, J-cost functional equation uniqueness, AlexanderDuality for D=3, phi-ladder constants) contains no theorems about polynomial system solving, homotopy continuation, or compositional decomposition of algebraic maps. Applications to reflection-group invariants are mentioned but do not invoke RS cost functions, 8-tick periodicity, or parameter-free constant derivations. Hence orthogonal.","tokens_in":58512,"confidence":"high","tokens_out":212,"duration_ms":18906,"cache_read_input_tokens":32896,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"A probabilistic algorithm solves composable polynomial systems by reducing them to an equivalent system in the inner variables, with complexity polynomial in input size and number of solutions.","keywords":["composable polynomial systems","symbolic homotopy","isolated regular solutions","probabilistic algorithm","finite reflection groups","invariant polynomials","Chevalley-Shephard-Todd theorem","algebraic independence"],"falsifier":"Running the algorithm on an explicit composable system with known isolated regular solutions and observing either that it misses some solutions or that its running time grows faster than polynomial in the input size and the number of solutions.","tokens_in":2634,"feed_emoji":"📐","tokens_out":776,"duration_ms":34262,"temperature":0.7,"pith_summary":"The paper focuses on polynomial systems where each equation is a composition of an outer polynomial with a shared set of inner polynomials. It shows that this structure permits reducing the original system in n variables to a simpler system involving only those inner polynomials. A probabilistic algorithm is then used to compute all isolated regular solutions of the reduced system. A sympathetic reader would care because many systems arising in invariant theory or algebraic computations possess this structure, making direct solution methods impractical due to high degrees while the reduced version stays manageable. The approach targets cases such as polynomials lying in a subring generated by algebraically independent elements or invariant rings of finite reflection groups.","feed_headline":"Algorithm reduces composable polynomials to polynomial-time solving","feed_subtitle":"By rewriting each equation as a composition with shared inner polynomials, the method computes all isolated regular solutions with cost that","key_machinery":"The composable structure, expressed as each f_i = h_i(g_1, …, g_n), which allows reduction of the full system to an equivalent system in the g variables before applying symbolic solution methods.","core_discovery":"We study the problem of computing the isolated regular solutions of a system of n polynomial equations in n variables over a field of characteristic zero. For systems with composable structure, where each polynomial f_i equals h_i of the inner polynomials g_1 through g_n, we reduce the original system to one in the g_j variables. We present a probabilistic algorithm that computes all isolated regular solutions, with arithmetic complexity polynomial in the input size and in the number of solutions. Applications include systems belonging to the subring generated by algebraically independent g_j and systems of invariant polynomials under finite reflection groups such as symmetric groups, hyperc","pith_inferences":["The same reduction idea could apply to other structured systems in algebraic geometry that exhibit similar functional composition without being explicitly labeled as composable.","If the inner polynomials are algebraically dependent, the reduction step would require additional handling of relations among the g_j, potentially preserving polynomial complexity only under extra conditions.","Implementation in computer algebra systems could make previously intractable invariant computations routine for larger reflection groups."],"forward_implications":["The method directly applies to any system whose polynomials lie in the subring generated by a set of algebraically independent polynomials g_1 to g_n.","It solves systems of invariant polynomials under finite reflection groups because the Chevalley-Shephard-Todd theorem guarantees that the invariant ring is a polynomial algebra.","The reduction improves the efficiency of symbolic homotopy or other algebraic solvers for any system possessing the stated composable form.","The probabilistic nature of the algorithm yields all isolated regular solutions with high probability while keeping complexity polynomial."],"fun_headline_variants":["Symbolic homotopy for composable polynomial systems","Reducing composable polynomials cuts solving complexity","Polynomial complexity method for composable equations","Symbolic solver exploits composable polynomial structure","Algorithm handles invariants under reflection groups efficiently"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The given polynomials must admit a composable structure in which every equation is a composition using the same collection of inner polynomials.","fun_headline_variants_meta":{"raw":{"variants":["Symbolic homotopy for composable polynomial systems","Reducing composable polynomials cuts solving complexity","Polynomial complexity method for composable equations","Symbolic solver exploits composable polynomial structure","Algorithm handles invariants under reflection groups efficiently"]},"model":"grok-4.3","cost_usd":0.007781,"raw_usage":{"total_tokens":3614,"prompt_tokens":788,"num_sources_used":0,"completion_tokens":52,"cost_in_usd_ticks":77812000,"prompt_tokens_details":{"text_tokens":788,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2774,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":788,"tokens_out":52,"duration_ms":51181,"temperature":1.0,"reasoning_tokens":2774,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-22T01:35:33.562264+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Running the algorithm on an explicit composable system with known isolated regular solutions and observing either that it misses some solutions or that its running time grows faster than polynomial in the input size and the number of solutions.","supporting_citations":[],"review_version":1}