{"id":"def98043-7587-46ea-9c85-46562209c0b8","arxiv_id":"2505.09709","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For a toric ideal I_A, the t-th symbolic power equals the kernel of a map sending each monomial to the t-fold tensor power of its exponent vector, and also equals I_A^t saturated by the product of all variables.","lead":"This paper gives two new ways to describe and compute the symbolic powers of toric ideals: as kernels of certain tensor-valued linear maps, and as saturations of ordinary powers by the product of all variables. For toric ideals, this makes symbolic powers computable from a small generating set of the lattice, which can be much faster than previous methods.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.9 is false as stated: for A=(1,1), 2e1-e1^2+2e2-e2^2 lies in ker π(1) but not in I_A^(2), so the claimed kernel characterization needs correction.","rationale":"The reader correctly identifies a genuine error in Lemma 3.7: the second expansion is not symmetrized, a fact visible already for t=2 and s=1. However, that mistake is not the most load-bearing failure. Using the first, correct part of Lemma 3.7, the coefficient of each elementary tensor in π(t-1)(m^a f) is exactly the symmetrized sum needed to prove π(0)(G_P)=0, so Theorem 3.8's intended argument can be repaired without changing the statement. The more serious problem is Theorem 2.9, which is false as written: the equality I_A^(t+1)=ker(π(t)) fails even for the minimal toric ideal (e1-e2) with A=(1,1). The counterexample is not exotic; it is a direct consequence of the definition of π(t). The proof's reverse inclusion implicitly assumes every element of ker(π(t)) is φ_A-homogeneous, but that is not true and Lemma 2.8 cannot be applied to non-homogeneous elements. The paper's computational framework can still be saved by replacing the false set equality with the statement that I_A^(t+1) is generated by the φ_A-homogeneous kernel components, namely ⊕_σ ker(π(t)|_{VA,σ}). Since this corrected version appears to hold and supports the saturation theorem, the appropriate verdict is conditional revision rather than outright rejection.","tokens_in":1125,"tokens_out":1063,"duration_ms":294692,"concrete_test":"Run Macaulay2 (or a direct computation): R=QQ[e1,e2]; phi=map(QQ[x], R, {x,x}); I=ker phi=ideal(e1-e2); f=2*e1-e1^2+2*e2-e2^2. Check π(1)(f)=2*(1,0)-(2,0)+2*(0,1)-(0,2)=(0,0), while ∂f/∂e1=2-2*e1 is not in I, so by Nagata-Zariski f is not in I^(2). This confirms Theorem 2.9 as stated is false. Then verify the corrected fiberwise statement I^(2)=ideal(e1^2-2*e1*e2+e2^2) still holds.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Section 2's headline characterization is false. Take A=(1,1), so φ(e1)=φ(e2)=x, IA=(e1-e2), and I_A^(2)=(e1-e2)^2. The polynomial f=2e1-e1^2+2e2-e2^2 satisfies π(1)(f)=2(1,0)-(2,0)+2(0,1)-(0,2)=0, so f∈ker(π(1)). But ∂f/∂e1=2-2e1∉(e1-e2), so by the Nagata-Zariski theorem f∉I_A^(2). Hence ker(π(1)) strictly contains the direct sum ⊕_σ ker(π(1)|_{VA,σ}), and the set-theoretic equality I_A^(2)=ker(π(1)) fails. The proof's reverse inclusion treats an arbitrary element of ker(π(t)) as if it were φ_A-homogeneous and invokes Lemma 2.8, which only applies to elements of a single fiber VA,σ. The correct statement is that I_A^(t+1) is the ideal generated by the φ_A-homogeneous kernel components, not the full kernel. This does not by itself disprove Theorem 3.8, because that proof uses only the homogeneous inclusion I_A^(t)⊆ker(π(t-1)), but the paper as written states a false central theorem. Separately, the reader's concern about Lemma 3.7 is real: its second formula omits symmetrization, so π(2)(f_u^2)=2u⊗u, not u⊗u; however, that error is repairable because the first, symmetrized formula yields the coefficient identities needed in Theorem 3.8.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies symbolic powers of toric ideals. It defines linear maps π(t) on the polynomial ring R=K[e_1,...,e_n] by π(t)(e^α)=α^{⊗t}, and claims in Theorem 2.9 that I_A^{(t+1)}=ker(π(t)) for the toric ideal I_A. In Section 3, the authors use this description to prove Theorem 3.5 for the second symbolic power and Theorem 3.8 for all t, asserting that I_A^{(t)}=I_A^t:m^∞, where m=e_1⋯e_n. A final remark proposes using a lattice-basis ideal J in place of I_A to speed up computations of symbolic powers. The intended results are structurally attractive and computationally relevant, but the proof as written contains a false theorem and an internally inconsistent lemma.","tokens_in":20276,"tokens_out":18518,"duration_ms":175312,"significance":"If the saturation statement I_A^{(t)}=I_A^t:m^∞ is correct, it is a clean and useful description of symbolic powers of toric ideals, and Remark 3.10 gives a concrete algorithmic benefit: one can saturate powers of a smaller lattice-basis ideal instead of the full toric ideal. The kernel description, once corrected to hold for φ_A-homogeneous components, also provides an explicit linear-algebra method for testing membership in symbolic powers. The paper is honest about its computational inspirations and uses standard external benchmarks (Sturmfels' Lemma 12.2 and the Nagata-Zariski theorem) rather than ad hoc assumptions. However, both main theorems are affected by the errors detailed below, so the manuscript needs substantial revision before the claims can be accepted.","major_comments":[{"comment":"The equality I_A^{(t+1)} = ker(π(t)) is false as stated. For A=(1,1), I_A=(e_1-e_2) and t=1, the polynomial f=2e_1-e_1^2+2e_2-e_2^2 satisfies π(1)(f)=0, but f∉I_A^{(2)}=(e_1-e_2)^2; for instance ∂f/∂e_1=2-2e_1∉(e_1-e_2), so the Nagata-Zariski criterion excludes f. The failure is structural: an element of ker(π(t)) need not be φ_A-homogeneous, and the reverse-inclusion proof in Theorem 2.9 applies Lemma 2.8 to the whole element f, although Lemma 2.8 is only valid on a fixed fiber V_{A,σ}. The apparently intended statement is I_A^{(t+1)} = ⊕_σ ker(π(t)|_{V_{A,σ}}), the direct sum over φ_A-homogeneous components. This correction is compatible with the rest of the paper because Theorem 3.8 only needs the inclusion I_A^{(t)}⊆ker(π(t-1)), but Theorem 2.9 as written is false and must be repaired.","section":"§2, Theorem 2.9"},{"comment":"The second displayed formula in Lemma 3.7 is inconsistent with the first. Take t=2, s=1, and v_1=v_2=u_1. The first formula gives π(2)(f_{u_1}^2)=2u_1⊗u_1, since the two permutations contribute the same tensor, while the second formula gives u_1⊗u_1. The missing factor is the sum over Sym(2); in general the coefficient of u_{p_1}⊗⋯⊗u_{p_t} should be Σ_{ω∈Sym(t)} ∏_{i=1}^t λ_{p_i,ω(i)}, not ∏_{i=1}^t λ_{p_i,i}. Theorem 3.8 uses the uncorrected identity to conclude coefficient-by-coefficient vanishing of Σ_j c_j ∏_i λ^{(j)}_{p_i,i}. With the symmetrized coefficient, linear independence of the tensors gives Σ_j c_j Σ_{p'∈Perm(p)} ∏_i λ^{(j)}_{p'_i,i}=0, which is precisely what is needed for the G_P computation after summing over permutations. Thus the gap is repairable, but the lemma and its application in Theorem 3.8 are incorrect as written.","section":"§3, Lemma 3.7"},{"comment":"The proof of Theorem 3.8 relies on both defective results. It invokes Theorem 2.9 to conclude that π(t-1)(m^a f)=0 from m^a f∈I_A^{(t)}, and it invokes Lemma 3.7 to pass from this vanishing to the coefficient identities that ultimately show G_P∈I_A. Since Theorem 2.9 is false as stated and Lemma 3.7 contains the erroneous factor, the central saturation claim I_A^{(t)}=I_A^t:m^∞ is not established by the current text. The intended argument appears salvageable: the needed direction of Theorem 2.9 is only the inclusion I_A^{(t)}⊆ker(π(t-1)), and the corrected symmetrized version of Lemma 3.7 yields the coefficient identities after summing over permutations. The authors should restate and prove these two ingredients correctly before the saturation theorem can be accepted.","section":"§3, Theorem 3.8"}],"minor_comments":[{"comment":"The proof of Lemma 2.4 contains an erroneous step: after writing g_i=B_1+⋯+B_N+C_1+⋯+C_M, the text claims that because C_k∉V_{A,σ} 'we must have that N is even and that Σ φ_A(B_j)=0'. In fact Σ φ_A(B_j)=N x^σ, which is nonzero over characteristic zero unless N=0. The statement of the lemma is nevertheless true, because every element of I_A is already φ_A-homogeneous: monomials with different φ_A-degree map to distinct monomials x^σ and cannot cancel. The proof should be replaced by a correct argument.","section":"§2, Lemma 2.4"},{"comment":"The manuscript contains numerous typographical and spacing errors, e.g. 'Then th regular poweris defined as In =I·I···I| {z } n' and inconsistent use of I_A^{(t)} versus I^{(t)}. A careful proofreading pass is needed.","section":"Throughout"},{"comment":"The notation π(s)(e^α)=α⊗⋯⊗α overloads the symbol α: it is an exponent vector in N^n and simultaneously used as a vector in K^n. This should be clarified, especially because Lemma 3.7 later applies π(t) to integer vectors v_i that may have negative entries.","section":"Definition 2.5"},{"comment":"The Macaulay2 code in Example 3.11 is not self-contained: the function 'spaghetti' refers to variables that are not defined in the displayed code. It would be helpful to move the full working code to an ancillary file or to include the missing definitions.","section":"Example 3.11"}],"recommendation":"major_revision","confidential_remarks":"The paper is not ready for publication in its current form: Theorem 2.9 is false as stated and Lemma 3.7 is internally inconsistent. However, the central saturation statement is plausibly true and both errors appear repairable within the manuscript's scope, so I recommend major revision rather than rejection. I saw no problems with citation practice or with the external benchmarks used."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nWorth knowing about, but not in current form. The headline result — symbolic powers of a toric ideal satisfy I_A^{(t)} = I_A^t : m^∞ where m is the product of all variables — is plausible and likely correct. The computational reduction in Remark 3.10 to J^t : m^∞ for a lattice-basis ideal J is a genuinely useful contribution. The kernel-of-π(t) point of view is also a nice way to compute symbolic powers in small cases.\n\nThe trouble is that the paper as written contains two concrete mathematical errors. Theorem 2.9, claiming I_A^{(t+1)} = ker(π(t)) = ⊕_σ ker(π(t)|_{V_{A,σ}}), is false. Take A=(1,1), so I_A=(e1-e2). The polynomial f=2e1-e1^2+2e2-e2^2 lies in ker(π(1)) because 2(1,0)-(2,0)+2(0,1)-(0,2)=0. But f is not in I_A^{(2)}=(e1-e2)^2: ∂f/∂e1 = 2-2e1, which is not a multiple of e1-e2. The proof applies Lemma 2.8 outside its domain (single fibers) to conclude the reverse inclusion, which is invalid. The correct statement would be that I_A^{(t+1)} is generated by the φ_A-homogeneous pieces in the kernel, not the full kernel.\n\nLemma 3.7 also has a false second formula. For s=1, t=2, v1=v2=u, π(2)(f_u^2) = 2u⊗u, not u⊗u. The proof of Theorem 3.8 relies on this coefficient formula. The error is repairable — using the symmetrized first part of the lemma recovers the needed identities (permanents), so the saturation theorem likely survives. But as written, the proof of the paper's central result is not valid.\n\nThe good news is that the left-to-right inclusion in Theorem 2.9, the t=2 saturation theorem, and the computational method are all sound in substance. The citations are fine and the derivations are transparent, with no fitted parameters.\n\nI would send this to a referee, because the saturation formula is central and the computational reduction is valuable, and both look salvageable. But the referee should require a corrected Theorem 2.9 and corrected Lemma 3.7 before acceptance. As it stands, it needs serious revision.","headline":"Main saturation formula is plausible, but the paper as written has a false main theorem (Theorem 2.9) and a false technical lemma (Lemma 3.7); needs repair before acceptance.","tokens_in":20845,"tokens_out":11405,"would_cite":false,"duration_ms":98989,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13D02","13P10","05E40"],"pacs":[],"model":"deepseek-v4-flash","headline":"For toric ideals, symbolic powers are saturations of ordinary powers.","keywords":["toric ideals","symbolic powers","saturation","kernel of linear maps","binomial ideals","lattice ideals","regular powers"],"falsifier":"Take t = 2, let the lattice be spanned by a single vector u_1, and set v_1 = v_2 = u_1. Lemma 3.7 predicts \\$pi^{{(2)}}$(f_{u_1}^2) = u_1 \\otimes u_1, but expanding f_{u_1}^2 = $e^{{2u_1}}$ - $2e^{{u_1}}$ + 1 gives 2 u_1 \\otimes u_1. This one computation settles whether the lemma, and with it the proof of the saturation theorem as written, can be correct.","tokens_in":19664,"feed_emoji":"🔢","tokens_out":6988,"duration_ms":60021,"temperature":0.7,"pith_summary":"This paper argues that the symbolic powers of a toric ideal are far more tractable than the general theory suggests: the t-th symbolic power $I_A^{{(t)}}$ equals the ordinary t-th power I_A^t saturated by the monomial m = e_1...e_n, the product of all variables. The authors also characterize $I_A^{{(t+1)}}$ as the kernel of an explicit linear map that sends each monomial e^\\$\\alpha$ to the t-fold tensor power \\$\\alpha$ \\otimes \\cdots \\otimes \\$\\alpha$. This matters because symbolic powers carry geometric information, and the usual route to them requires computing a difficult auxiliary ideal; here a single saturation plus linear algebra suffices.","feed_headline":"Toric symbolic powers are just saturations of regular powers","feed_subtitle":"One multiplication by all variables turns the t-th power into the t-th symbolic power, making computations tractable.","key_machinery":"The load-bearing object is the linear map \\$pi^{{(t)}}$ : K[e_1,\\ldots,e_n] \\to (K^n)^{\\otimes t} defined on monomials by \\$pi^{{(t)}}$(e^\\$\\alpha$) = \\$\\alpha$ \\otimes \\cdots \\otimes \\$\\alpha$, together with the binomial forms f_u = $e^{{u^+}}$ - $e^{{u^-}}$ for lattice vectors u \\in \\ker(A). Lemma 3.7 computes \\$pi^{{(t)}}$ of a product of t such binomials as a signed sum of tensor products of the lattice vectors, with coefficients given by the coordinates of the vectors in a basis of \\ker(A). That expansion is what lets the authors move from membership in the symbolic power, detected by the kernel of \\$pi^{{(t)}}$, to membership in a regular power after multiplying by a suitable power of m.","core_discovery":"The central claim is that for every toric ideal I_A in K[e_1, ..., e_n], the t-th symbolic power is obtained by saturating the ordinary power by the product of all variables: $I_A^{{(t)}}$ = I_A^t : m^\\infty. The proof proceeds through a second description: $I_A^{{(t+1)}}$ is exactly the kernel of the linear map \\$pi^{{(t)}}$ that sends e^\\$\\alpha$ to \\$alpha^{{\\otimes t}}$, restricted to the finite-dimensional fiber spaces V_{A,\\$\\sigma$}. The two descriptions are connected by expanding products of binomials f_{v_i} = $e^{{v_i^+}}$ - $e^{{v_i^-}}$ attached to lattice vectors v_i \\in \\ker(A), then applying a classical binomial-saturation lemma to pass from a basis of the lattice to the whole toric ideal.","pith_inferences":["If the saturation formula I^{(t)} = I^t : m^\\infty extends beyond toric ideals to other lattice ideals or binomial ideals, the same computational shortcut would apply there; the paper does not claim this extension.","The kernel description suggests that the gap between I^t and I^{(t)} is governed by tensor decomposability: elements of the symbolic power correspond to linear dependencies among tensor powers of exponent vectors, a viewpoint that may connect to problems on symmetric tensor rank.","The coefficient expansion in Lemma 3.7, if correct, would supply explicit binomial generators for I_A^{(t)} from a lattice basis; checking the expansion for small t is a direct calculation any reader can perform."],"forward_implications":["For every toric ideal I_A, the equality I_A^{(t)} = I_A^t : m^\\infty holds, so symbolic powers can be computed by one saturation with the all-variable monomial.","The t-th symbolic power is the kernel of an explicit linear map on each fiber, yielding a matrix description that can be used directly in computation.","Combining the saturation equality with the classical lattice-basis lemma, I_A^{(t)} = J^t : m^\\infty for J generated by a lattice basis, which is usually a much smaller ideal than I_A itself.","The methods detect elements of symbolic powers that have degree lower than the minimal degree of the ordinary power; an example for the ideal of the complete graph K_5 exhibits a degree-5 element in I^{(3)} that is not in I^3."],"supporting_citations":[{"why":"Supplies the binomial-saturation lemma: saturating the ideal generated by a lattice basis by m recovers the full toric ideal.","marker":"[23]"},{"why":"Gives the Nagata-Zariski theorem that identifies symbolic powers through differential operators, used as the entry point of Section 2.","marker":"[9]"},{"why":"Provides the prime-ideal characterization of symbolic powers as {f : fg in I^t for some g not in I}, used to prove one containment in the saturation equality.","marker":"[18]"},{"why":"Cited for the same prime-ideal equivalence in the proof of the main theorem.","marker":"[17]"}],"fun_headline_variants":["Symbolic powers of toric ideals are saturations by product of variables","Toric symbolic powers: saturate the ordinary power by all variables","For toric ideals, symbolic powers come from saturating regular powers","Saturate by product of variables to get toric symbolic powers","Symbolic powers of toric ideals: a single saturation formula"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of the main theorem leans entirely on Lemma 3.7's coefficient formula for the tensor image of a product of binomials; if that formula fails, the chain connecting symbolic powers to saturations breaks down.","fun_headline_variants_meta":{"raw":{"variants":["Symbolic powers of toric ideals are saturations by product of variables","Toric symbolic powers: saturate the ordinary power by all variables","For toric ideals, symbolic powers come from saturating regular powers","Saturate by product of variables to get toric symbolic powers","Symbolic powers of toric ideals: a single saturation formula"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000303,"raw_usage":{"total_tokens":1657,"prompt_tokens":774,"completion_tokens":883,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":390,"completion_tokens_details":{"reasoning_tokens":791}},"tokens_in":390,"tokens_out":883,"duration_ms":8006,"temperature":1.0,"reasoning_tokens":791,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:28:47.798402+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take t = 2, let the lattice be spanned by a single vector u_1, and set v_1 = v_2 = u_1. Lemma 3.7 predicts \\$pi^{{(2)}}$(f_{u_1}^2) = u_1 \\otimes u_1, but expanding f_{u_1}^2 = $e^{{2u_1}}$ - $2e^{{u_1}}$ + 1 gives 2 u_1 \\otimes u_1. This one computation settles whether the lemma, and with it the proof of the saturation theorem as written, can be correct.","supporting_citations":[{"cited_title":"Sturmfels.Grobner bases and convex polytopes, volume 8","cited_arxiv_id":null,"evidence_quote":"Supplies the binomial-saturation lemma: saturating the ideal generated by a lattice basis by m recovers the full toric ideal."},{"cited_title":"Eisenbud.Commutative algebra: with a view toward algebraic geometry, volume 150 ofGraduate Texts in Mathematics","cited_arxiv_id":null,"evidence_quote":"Gives the Nagata-Zariski theorem that identifies symbolic powers through differential operators, used as the entry point of Section 2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the prime-ideal characterization of symbolic powers as {f : fg in I^t for some g not in I}, used to prove one containment in the saturation equality."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Cited for the same prime-ideal equivalence in the proof of the main theorem."}],"review_version":1}