Symbolic Powers of Toric Ideals

Giuseppe Favacchio; Graham Keiper

REVIEW 3 major objections 4 minor 24 references

Symbolic Powers of Toric Ideals

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read For toric ideals, symbolic powers are saturations of ordinary powers.

desk verdict 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. read the letter →

arxiv 2505.09709 v1 pith:JLDG2ARU submitted 2025-05-14 math.AC math.AG

Giuseppe Favacchio , Graham Keiper This is my paper · ORCID

classification math.ACmath.AG MSC 13D0213P1005E40

keywords toricidealssymbolicpowerssaturationkerneloflinearmapsbinomiallatticeregular

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

Worth 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.

The 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.

Lemma 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.

The 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.

I 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.

Referee Report

3 major / 4 minor

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.

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 (3)

[§2, Theorem 2.9] 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.
[§3, Lemma 3.7] 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.
[§3, Theorem 3.8] 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.

minor comments (4)

[§2, Lemma 2.4] 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.
[Throughout] 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.
[Definition 2.5] 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.
[Example 3.11] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central claims are derived from external theorems; the noted defects are correctness issues, not circular reasoning.

full rationale

The paper's derivation chain is anchored to external results: the Nagata-Zariski theorem (quoted in Section 1), Sturmfels's Lemma 12.2 (Lemma 3.1), and Grifo's characterization of symbolic powers for prime ideals. The kernel characterization (Theorem 2.9) is derived from Nagata-Zariski and Lemma 2.3 by induction; the saturation results (Theorems 3.5 and 3.8) are derived from Theorem 2.9, Lemma 3.1, and algebraic manipulation of binomials, rather than being assumed or fitted. No parameter is fitted and no quantity is renamed as a prediction. The authors' self-citations are contextual and do not carry any load-bearing step; the saturation multiplier is not imported from author-specific prior work. The paper does contain serious correctness defects: Lemma 3.7's second formula omits the necessary symmetrization, and the reverse inclusion in Theorem 2.9 applies fiber-restricted arguments to arbitrary kernel elements. However, these are soundness errors, not circularity: the false lemma is an internal assumption, not a restatement of the target theorem, and the target theorem is not used to justify the lemma. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claims rest on standard theorems in commutative algebra and on the structural properties of toric ideals. No free parameters are fitted. The only potentially fragile input is the unproven assumption about the associated primes of powers of a toric ideal, which the flawed proof attempted to bypass.

assumptions (4)

standard math Nagata-Zariski theorem: for a prime ideal I, I^(n) = {f | ∂^k f/∂M ∈ I for all monomials M and k<n}.
Invoked in Section 2 as the basis for the kernel characterization and in Lemma 2.4.
standard math Sturmfels Lemma 12.2: if C spans ker(A), then (J_C : m^∞) = I_A.
Used as Lemma 3.1 to relate a lattice basis ideal to the toric ideal, and again in Remark 3.10.
domain assumption The polynomial ring is over an algebraically closed field of characteristic zero.
Required for the stated Nagata-Zariski theorem and for the derivative arguments.
domain assumption All associated primes of I_A^t other than I_A contain a variable, so saturation by m isolates the I_A-primary component.
This is implicitly used in the intended structure of Theorem 3.8, though the paper attempts a direct proof that is compromised by the false Lemma 3.7.

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Cite this review

Pith. "Pith review of Symbolic Powers of Toric Ideals." pith.science (2026). https://pith.science/paper/JLDG2ARU

@misc{pith2026250509709,
  author       = {Pith},
  title        = {Pith review of: Symbolic Powers of Toric Ideals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JLDG2ARU}},
  note         = {Machine review of arXiv:2505.09709}
}

read the original abstract

This paper investigates the symbolic powers of toric ideals. We first describe them in terms of the kernel of certain linear maps derived from the lattice structure of the toric ideal. Furthermore, we apply our results to show that symbolic powers of a toric ideal can also be expressed as saturations of regular powers with the monomial given by the product of all the variables. Finally, we conclude with a computationally significant result for computing symbolic powers of toric ideals.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Accepted for publication

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Abbott, A

J. Abbott, A. M. Bigatti, and L. Robbiano. CoCoA: a system for doing Computations in Commutative Algebra. Available athttp://cocoa.dima.unige.it

work page

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Bhaskara and A

K. Bhaskara and A. Van Tuyl. Comparing invariants of toric ideals of bipartite graphs.Proceedings of the American Mathematical Society, Series B, 10(19):219–232, 2023

work page 2023

[3] [3]

Bocci, S

C. Bocci, S. Cooper, E. Guardo, B. Harbourne, M. Janssen, U. Nagel, A. Seceleanu, A. V. Tuyl, and T. Vu. The Waldschmidt constant for squarefree monomial ideals.Journal of Algebraic Combinatorics, 44:875–904, 2016

work page 2016

[4] [4]

Bocci and B

C. Bocci and B. Harbourne. Comparing powers and symbolic powers of ideals.Journal of Algebraic Geometry, 19(3):399–417, 2010

work page 2010

[5] [5]

Carlini, H

E. Carlini, H. T. H` a, B. Harbourne, and A. Van Tuyl.Ideals of powers and powers of ideals. Springer, 2020

work page 2020

[6] [6]

Constantinescu and E

A. Constantinescu and E. Gorla. Gorenstein liaison for toric ideals of graphs.Journal of Algebra, 502:249–261, 2018

work page 2018

[7] [7]

S. M. Cooper, S. Da Silva, M. Gutkin, and T. Reimer. Splittings for symbolic powers of edge ideals of complete graphs. Journal of Commutative Algebra, 16(2):183–196, 2024

work page 2024

[8] [8]

Cummings, S

M. Cummings, S. Da Silva, J. Rajchgot, and A. Van Tuyl. Geometric vertex decomposition and liaison for toric ideals of graphs.Algebraic Combinatorics, 6(4):965–997, 2023. SYMBOLIC POWERS OF TORIC IDEALS 17

work page 2023