{"id":"891f2892-a6c9-4639-8180-510bd34aa2f9","arxiv_id":"1908.00899","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"An algorithm and supporting toolkit for numerical irreducible decomposition of multiprojective varieties, reducing the computation to curves in affine space and generalizing prior work from two to arbitrarily many projective factors.","lead":"This paper develops numerical methods for working with algebraic varieties defined by multihomogeneous polynomial systems, using finite sample points called witness collections. The new algorithms compute dimensions, test membership, and decompose such varieties into irreducible pieces, which is useful for solving polynomial systems in kinematics and robotics.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Algorithm 6.2 Step (7) as written is dimensionally inconsistent: L := Le ∪ {ℓ_j} contains 2|e|-1 linear forms, so CL = Xp ∩ V(L) has dimension 1-|e|, not a curve; the claimed reduction to an affine curve fails as printed.","rationale":"The paper's toolkit in Sections 1–5 is largely well-founded: the support/multidimension theory, product detection, refining/coarsening homotopies, and slicing Lemma 5.1 are presented coherently and the examples are consistent with the stated multidegrees. The central claim, however, hangs on Algorithm 6.2, which is supposed to reduce numerical irreducible decomposition to the case of a curve in affine space. The reader flagged the heuristic nature of the trace test and Algorithm 2.5's lack of a stopping criterion; those are real and acknowledged limitations. My independent check found a more elementary defect: Step (7) as printed builds a system with 2|e|−1 linear forms, which cannot cut an irreducible variety down to a curve. The dimension count is off by |e|, and the paper's own Example 6.4 would produce an empty set under the literal instruction. The intended construction is recoverable from the prose and the proof of correctness — replace Le by Lm and add the missing witness slice ℓ — and the corrected version would likely work modulo the usual numerical heuristics. Because the flaw is localized and fixable, REJECT would be too harsh, but the manuscript as submitted cannot substantiate its headline algorithm. A CONDITIONAL decision requiring correction of Algorithm 6.2 (and a clarification of the heuristic completeness assumptions) is appropriate.","tokens_in":25808,"tokens_out":20485,"duration_ms":207162,"concrete_test":"Implement Algorithm 6.2 Step (7) literally on Example 6.4: Y = V(f,h) ⊂ C_x×C_y×C_z×C_w, with f = 1+2x+3y^2+4z^3+5w^4 and h = 1+2x+3y+5z+7w+11xy+13xz+17xw+19yz+23yw+29zw+31xyz+37xyw+41xzw+43yzw+47xyzw. Take m = (0,0,0,0), e = 1100, so Le consists of one general linear form in x and one in y, and ℓ_1 is a general linear form in (x,y). Form L = Le ∪ {ℓ_1} and run the homotopy (4.1) with system F = (f,h). The three independent linear equations on the 2-dimensional surface Y leave no bounded endpoints, so the routine returns an empty set. Re-run with L = Lm ∪ {ℓ_1}; the resulting curve has degree 15 and the trace test succeeds. This distinguishes the textual defect from a heuristic failure.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Let X' := Xp ∩ V(Lm), and choose e ∈ Dim(X') as in Step (4). Since e ∈ {0,1}^k and |e| = dim(X'), the collection Le defined in Section 1.3 consists of exactly |e| linear forms, one in each coordinate group with e_i = 1. Step (6) then adds |e|−1 further forms ℓ_j. Thus the L built in Step (7) has (|e|) + (|e|−1) = 2|e|−1 linear forms total. Intersecting X' with V(Le) already cuts the dimension from |e| to 0; the extra ℓ_j make Xp ∩ V(L) generically empty or of negative dimension, not a curve. Equivalently, CL has dimension 1−|e|, which is 1 only when |e|=0 — but then Step (6) would require −1 forms, so no case yields the asserted curve. Example 6.4 makes the contradiction concrete: Y = V(f,h) is a surface (dim X' = 2), and with e = 1100 the literal Step (7) gives three independent linear forms in C^4, so CL is empty, whereas the example reports an irreducible curve of degree 15. The surrounding text and proof indicate the intended L is Lm ∪ {ℓ_1,...,ℓ_{|e|−1}} (one form, degree 15), but as printed Algorithm 6.2 cannot be executed. This is an internal inconsistency in the central algorithm, independent of the heuristic trace-test stopping criterion.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a numerical toolkit for manipulating multiprojective (and multiaffine) varieties, extending prior work by the same authors. The main ingredients are: computing the (multi)dimension of an irreducible component from a smooth point via Jacobian ranks (Algorithm 2.3); completing a partial witness collection by monodromy (Algorithm 2.5); detecting Cartesian product structure from the polymatroid polytope of the multidimension (Theorem 3.1); transforming witness collections under refinement and coarsening of the multiprojective structure (Algorithms 4.3 and 4.5); and slicing with linear forms while controlling dimension, degree, and irreducibility (Lemma 5.1, Algorithm 5.3). These tools are assembled into Algorithm 6.2, which aims to partition an arbitrary finite set of general smooth points on a multiprojective variety according to the irreducible components, by reducing each component to an affine curve and using monodromy plus a trace test. The paper includes worked examples, including a Richardson variety, fiber-product systems for hyperboloid rulings, and exceptional planar pentads.","tokens_in":26129,"tokens_out":11608,"duration_ms":109176,"significance":"If correct, the paper would provide a practical and conceptually clean way to perform numerical irreducible decomposition for multiprojective varieties without passing to a large Segre embedding, extending the curve-based trace test of [10] to arbitrarily many projective factors. The geometric statements (Theorem 3.1, Lemma 5.1, Propositions 4.2) are proved against standard algebraic geometry, and the examples show genuine computational savings (e.g., 16 witness points versus 120 in the hyperboloid example, and 14,828 versus roughly 10^8 in the pentad example). The paper is transparent about the heuristic nature of the trace test and about the lack of a stopping criterion in Algorithm 2.5. However, the central decomposition algorithm contains a concrete dimensional inconsistency in its Step (7), and its completeness relies on unproven heuristics; these issues must be addressed before the algorithmic claims can be accepted.","major_comments":[{"comment":"Algorithm 6.2 invokes Algorithm 2.5 in Step (7) to compute a witness set for CL, but the proof of correctness of Algorithm 2.5 explicitly states that it \"does not have a stopping criterion, and is therefore technically not an algorithm.\" Moreover, the trace test is described in Section 1.2 as a \"heuristic stopping criterion for monodromy break up.\" Consequently, the partition output by Algorithm 6.2 is not certified: if monodromy fails to discover all points of CL, or if the trace test accepts an incomplete partition, the membership test in Step (8) may misclassify points, and the output N may merge or miss irreducible components. The paper should either supply a termination and completeness argument under explicit genericity assumptions or clearly label Algorithm 6.2 as a heuristic procedure, separate from the proven geometric statements in Sections 3–5.","section":"Section 2.2 and Section 6.2"}],"minor_comments":[{"comment":"The notation \"Y /integerdivideV(Λ)\" appears several times and should be the set difference \"Y \\ V(Λ)\" or \"Y ∖ V(Λ)\"; the current rendering obscures the intended meaning.","section":"Sections 4.1 and 4.2"},{"comment":"There is a typo: \"irredicubility\" should be \"irreducibility.\" Also, \"coordinatewise maximal integer vector\" should be defined explicitly (e.g., maximal with respect to the product order on [n•]).","section":"Section 6.2, Step (3)"},{"comment":"The symbol e is used both for the intrinsic dimension of X and for an element of [n•], leading to the confusing expression {e ∈ [n•] | |e| = e}. Please rename one of them (e.g., use d for the intrinsic dimension).","section":"Section 2.1, Equation (2.1)"},{"comment":"The line \"u0 + u1 + u2 + u4 = u0 + u1 + u2 + u4 = v0 + u1 + u3 + v4 = v0 + u1 + u3 + v4 = 0\" is ambiguous; it should be written as four separate equations so that the distinct variables u and u, and v0 and v0, are clearly handled.","section":"Section 7.2"}],"recommendation":"major_revision","confidential_remarks":"The paper's reliance on prior work by the same authors ([6], [10]) is substantial but appropriately acknowledged, and the new geometric statements are proved against independent facts. The main obstruction is the dimensional error in Algorithm 6.2 Step (7), which is central to the paper's algorithmic claim. The heuristic nature of the stopping criteria is a known feature of numerical algebraic geometry, but the paper should be explicit about the status of Algorithm 6.2 as a heuristic rather than a certified algorithm."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Zsolt, here's the read on arXiv:1908.00899. The toolkit is real: Theorem 3.1 (product detection via the support polytope) is clean, the refining/coarsening transformations are useful, and Lemma 5.1 gives a simple slicing rule with a concrete bookkeeping algorithm. The pentad example is the strongest advertisement: roughly 15,000 witness points instead of 10^8. That's a genuine payoff for respecting multiprojective structure.\n\nThe soft spots are real too. Algorithm 6.2 is a sketch, and it contains a concrete bug. Step (7) defines L := Le ∪ {ℓ_1,...,ℓ_{|e|-1}}, but Le already has |e| forms, so L has 2|e|-1 forms. If Xp ∩ V(Lm) has dimension |e|, that overslices to dimension 1 - |e|, not a curve. The surrounding text and Example 6.4 make clear the intended L is Lm (or the m-forms) plus the |e|-1 extra forms, so the fix is a one-line change. But as printed, Algorithm 6.2 cannot be executed. The proof of correctness never addresses the count, which is why this slipped through.\n\nBeyond the bug, the completeness of the decomposition rests on the trace test, which the paper itself labels heuristic, and on Algorithm 2.5, which lacks a termination criterion. The authors are transparent about both. That's to their credit, but it means the paper delivers a toolkit plus a plausible decomposition strategy, not a certified algorithm. Remark 6.3 also admits that coarsening generally loses completeness, so the one-to-one homotopy of the two-factor case does not generalize.\n\nI'd send it to review. The geometric core is sound, the examples are consistent, and the bug is fixable. A referee should require a corrected Step (7), a clearer statement of what the trace test is being trusted to do, and perhaps an explicit assumption under which the decomposition output is certified.","headline":"Solid multiprojective numerical-algebraic toolkit with a real application win, but the central decomposition algorithm has a dimensional bug in Step (7) and relies on acknowledged heuristics.","tokens_in":26648,"tokens_out":8036,"would_cite":true,"duration_ms":72291,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65H10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper develops a numerical toolkit that reduces irreducible decomposition of multiprojective varieties to monodromy and a trace test on an affine curve.","keywords":["numerical algebraic geometry","multiprojective variety","witness collection","numerical irreducible decomposition","monodromy","trace test","multihomogeneous polynomial","Segre embedding"],"falsifier":"Run Algorithm 6.2 on the hyperboloid fiber-product system of Section 7.1 starting from the 16-point witness set, and compare the two recovered ruling curves with the symbolic equations $\\{(\\lambda_1,\\lambda_2,-\\lambda_2,\\lambda_1): \\lambda_1^2+\\lambda_2^2=1\\}$ and $\\{(\\lambda_1,\\lambda_2,\\lambda_2,-\\lambda_1): \\lambda_1^2+\\lambda_2^2=1\\}$; any mismatch in the partition or in the recovered curve equations would show the reduction to curve trace test fails in a concrete instance.","tokens_in":25613,"feed_emoji":"📐","tokens_out":10631,"duration_ms":99483,"temperature":0.7,"pith_summary":"This paper is trying to make numerical algebraic geometry work directly on varieties inside a product of projective spaces, instead of forcing such a variety into a single projective space through the Segre embedding. The numerical description of a multiprojective variety is a \"witness collection\": for every multidegree vector at which a general linear section is finite, it records the section's isolated points. The authors build four operations on witness collections—computing the local multidimension, recognizing Cartesian product structure, refining and coarsening the factor structure, and slicing by a linear form—and assemble them into an algorithm that partitions any finite set of general smooth points by irreducible component. The payoff is that numerical irreducible decomposition of a multiprojective variety reduces to the standard case of a curve in affine space, where an efficient trace test applies. This generalizes the two-factor trace test to arbitrarily many projective factors.","feed_headline":"New toolkit cuts multiprojective decomposition down to curve case","feed_subtitle":"Witness collections, slicing, and coarsening turn any multiprojective variety into a curve for the trace test.","key_machinery":"The load-bearing object is the witness collection, the map $e \\mapsto (F, L_e, X \\cap V(L_e))$ over the support $\\operatorname{Dim}(X) \\subseteq \\prod_i \\{0,\\dots,n_i\\}$, where the support plays the role of dimension and the entries of the multidegree play the role of degree. The support is a polymatroid polytope, and Proposition 2.1 describes it by the inequalities $e_{i_1}+\\cdots+e_{i_s} \\le \\dim_I(X)$ for all subsets $I$ of factors, which Algorithm 2.3 computes from the rank of Jacobian submatrices. The action is carried by three mechanisms: monodromy, which completes a partial witness set by tracking points as the linear section moves in loops; the refining and coarsening homotopies of Section 4, which move between multilinear and bilinear sections and thereby change the multiprojective structure; and slicing, which replaces $\\operatorname{Dim}(X)$ by $\\{e-\\epsilon_i : e \\in \\operatorname{Dim}(X), e_i>0\\}$ and preserves (ir)reducibility when the $i$-th projection has dimension at least two. Algorithm 6.2 uses these mechanisms to pass from an arbitrary point to an affine curve whose witness set is complete, then applies the trace test.","core_discovery":"The central claim is that the extra structure of a multiprojective variety can be made algorithmic. A variety $X \\subset \\mathbb{P}^{n_1} \\times \\cdots \\times \\mathbb{P}^{n_k}$ is represented by a witness collection, which assigns to each $e \\in \\operatorname{Dim}(X)$ the isolated points of $X \\cap V(L_e)$, with $L_e$ a general linear section of multidegree $e$. The support $\\operatorname{Dim}(X)$ is characterized by projection inequalities (Proposition 2.1) and is computed locally from Jacobian ranks. From one general smooth point, monodromy completes a partial witness collection; the toolkit then recognizes when $\\operatorname{Dim}(X)$ factors as a product of polymatroid polytopes, which is exactly when $X$ is a Cartesian product, and it transforms witness collections under reparameterizations that refine or coarsen the product structure and under slicing by a linear form. Slicing preserves irreducibility whenever the projection to that factor has dimension at least two, so repeated slicing and coarsening reduce an irreducible component to an affine curve $\\mathcal{C}_L$; a complete witness set for that curve is obtained by monodromy with the trace test as stopping criterion, and that witness set yields a membership test that sorts all input points into components. This is the content of Algorithm 6.2.","pith_inferences":["Inference: the cost of the curve reduction depends on the order and type of slices and coarsenings chosen; the paper leaves the search for the minimal-degree reduction open, so an automatic strategy that minimizes the final curve degree would make the toolkit independently useful.","Inference: the local computation of $\\operatorname{Dim}(X)$ via projections is a numerical computation of the algebraic matroid of the variety, so the toolkit gives a practical way to extract matroid bases from a witness collection.","Inference: for fiber-product systems like the exceptional pentad, the same reduction could be used to certify exceptional components directly from one witness point, without ever forming the full witness collection in the high-dimensional ambient space.","Inference: a testable extension is to feed points from a long homotopy run into the membership test of Algorithm 6.2 incrementally, classifying them online instead of first collecting a full witness set."],"forward_implications":["For any multiprojective variety, numerical irreducible decomposition can be performed without computing the Segre embedding, avoiding the drastic growth in ambient dimension and degree that the embedding causes.","The trace test, previously limited to curves in products of two projective spaces, now serves as a stopping criterion for decomposing varieties with any number of projective factors.","A witness collection for a slice $X \\cap V(\\ell)$ is obtained from a witness collection for $X$ by shifting the support: $\\operatorname{Dim}$ loses the vectors with $e_i = 0$, and the multidegree is read off from the original collection.","When $\\operatorname{Dim}(X)$ factors as a product of polymatroid polytopes, $X$ itself is a Cartesian product and its witness collections factor, so membership can be tested in each factor separately.","The same machinery partitions arbitrary finite sets of general smooth points on $V(F)$, not just points that already lie in a witness collection."],"supporting_citations":[{"why":"establishes witness collections for multiprojective varieties and the fundamental algorithms this toolkit extends.","marker":"[6]"},{"why":"gives the trace test for curves in products of two projective spaces, which the decomposition algorithm generalizes.","marker":"[10]"},{"why":"characterizes the support of a multiprojective variety by projection inequalities, the basis for Algorithm 2.3.","marker":"[2]"},{"why":"introduces numerical irreducible decomposition for projective and affine varieties, the base problem being generalized.","marker":"[17]"},{"why":"supplies the monodromy breakup algorithm that partitions witness points according to irreducible components.","marker":"[18]"},{"why":"supplies the trace test used as the stopping criterion for monodromy breakup.","marker":"[19]"},{"why":"provides the monodromy solving method used to complete a partial witness collection from one point.","marker":"[3]"},{"why":"provides the fiber-product construction used in the hyperboloid and planar pentad examples.","marker":"[21]"}],"fun_headline_variants":["Witness collections turn multiprojective varieties into curves","Algorithmic toolkit slices multiprojective varieties to curves","Curve case suffices for multiprojective decomposition","Numerical toolkit reduces multiprojective decomposition to slicing","From multiprojective to curve: new toolkit for decomposition"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole decomposition rests on the assumption that monodromy, run with the trace test as a stopping rule, actually finds every point of the curve witness set and recognizes when a subset is a complete witness set; the paper itself notes that the completion step has no guaranteed stopping criterion and the trace test is a heuristic.","fun_headline_variants_meta":{"raw":{"variants":["Witness collections turn multiprojective varieties into curves","Algorithmic toolkit slices multiprojective varieties to curves","Curve case suffices for multiprojective decomposition","Numerical toolkit reduces multiprojective decomposition to slicing","From multiprojective to curve: new toolkit for decomposition"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000234,"raw_usage":{"total_tokens":1482,"prompt_tokens":916,"completion_tokens":566,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":532,"completion_tokens_details":{"reasoning_tokens":489}},"tokens_in":532,"tokens_out":566,"duration_ms":5491,"temperature":1.0,"reasoning_tokens":489,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:28:52.166935+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run Algorithm 6.2 on the hyperboloid fiber-product system of Section 7.1 starting from the 16-point witness set, and compare the two recovered ruling curves with the symbolic equations $\\{(\\lambda_1,\\lambda_2,-\\lambda_2,\\lambda_1): \\lambda_1^2+\\lambda_2^2=1\\}$ and $\\{(\\lambda_1,\\lambda_2,\\lambda_2,-\\lambda_1): \\lambda_1^2+\\lambda_2^2=1\\}$; any mismatch in the partition or in the recovered curve equations would show the reduction to curve trace test fails in a concrete instance.","supporting_citations":[{"cited_title":"Hauenstein and J.I","cited_arxiv_id":null,"evidence_quote":"establishes witness collections for multiprojective varieties and the fundamental algorithms this toolkit extends."},{"cited_title":"Leykin, J.I","cited_arxiv_id":null,"evidence_quote":"gives the trace test for curves in products of two projective spaces, which the decomposition algorithm generalizes."},{"cited_title":"Representable Chow classes of a product of projective spaces","cited_arxiv_id":"1612.00154","evidence_quote":"characterizes the support of a multiprojective variety by projection inequalities, the basis for Algorithm 2.3."},{"cited_title":"Sommese, J","cited_arxiv_id":null,"evidence_quote":"introduces numerical irreducible decomposition for projective and affine varieties, the base problem being generalized."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the monodromy breakup algorithm that partitions witness points according to irreducible components."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the trace test used as the stopping criterion for monodromy breakup."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the monodromy solving method used to complete a partial witness collection from one point."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the fiber-product construction used in the hyperboloid and planar pentad examples."}],"review_version":1}