The Mendez–Pinto–Villarreal conjecture in unmixed height three

Henry Zweiman

September 9, 2026

Abstract

Let be a monomial ideal whose minimal primary components are irreducible and whose associated primes all have height three. We prove that equality of all symbolic and ordinary powers forces the exponent of each variable to agree in every primary component containing it. Consequently is a standard variable weighting of its squarefree radical, which is itself Simis. This proves the unmixed height-three case of the Mendez–Pinto–Villarreal conjecture, without restrictions on the exponents or on the supports of minimal generators. The proof uses a classical complete-intersection criterion in dimension one to obtain rigidity on four variables. A putative disagreement then localizes to five variables and four support configurations. Three are excluded by symbolic-power witnesses or triangle specializations; the last admits an explicit obstructing vertex of the irreducible polyhedron for every choice of exponents.

1. Statement and context

Let , where is an arbitrary field. For an ideal its symbolic powers in this paper are

The ideal is Simis if for every integer . An irreducible monomial ideal is generated by positive powers of distinct variables. We assume throughout the main theorem that

where is a family of distinct three-element subsets of . Thus (1) is a minimal primary decomposition without embedded primes. Conversely, every monomial ideal with irreducible minimal primary components and all associated primes of height three has this form. This spells out the meaning of minimal irreducible decomposition used here: there is exactly one irreducible primary component for each associated prime.

For positive integers and a monomial ideal , denote by the ideal generated by the images of its monomials under . This is a standard variable weighting.

Theorem 1.

For an ideal as in (1), the following are equivalent:

1. is Simis.

2. For each , the integers are equal as ranges over the members of containing , and is Simis.

3. for a standard variable weighting , and is Simis.

Méndez, Vaz Pinto, and Villarreal [8], Conjecture 5.7, propose the implication from the Simis property to standard weighting for monomial ideals without embedded primes whose irreducible decomposition is minimal. Their Corollary 5.5 already proves invariance of the Simis property under standard weighting; the new implication in Theorem 1 is agreement of the primary-component exponents in unmixed height three.

The height-two case follows from Aslam [1], Theorem 2.3, and is also formulated as a case of the conjecture by Bijender and Kumar [2], Theorem 4.9. Bordoloi, Das, and Kumar [3,4] treat ideals constrained by the supports of their minimal generators; in particular [4], Theorem 3.10, treats support-three ideals, in the setup of that paper. Component height three and generator support three are different conditions. Our argument makes no assumption on generator support sizes or on their equality with the supports of the radical’s generators. The general conjecture remains outside the scope of this paper.

The main mechanism is local: two components containing a discrepantly weighted variable have union of size at most five. Four-variable localizations give a rigid complete-intersection structure. The remaining five-variable configurations can then be excluded uniformly in all positive exponents.

2. Monomial operations and dimension-one rigidity

We first record the operations used in the proof. In this section the sets in can have arbitrary sizes, but are distinct and incomparable under inclusion. Write . The power is -primary: membership of a monomial depends only on its exponents in . Localization at removes the other components. Consequently

For later use, the exact membership test is

Indeed, the right side says that pure-power factors from can be chosen to divide .

Lemma 2.

Suppose is Simis and has the decomposition just described.

1. For a set of variables, the ideal in obtained by retaining exactly the components with is Simis, if any components remain. Their exponents are unchanged.

2. Let set all other variables equal to zero. Suppose every is proper and nonzero, and their radicals are distinct and incomparable after deleting identical duplicate ideals. Then their intersection is Simis.

Proof.

For (1), invert the variables outside . A component disappears precisely when its support is not contained in . Localization commutes with powers and finite intersections, so the Simis equalities (2) pass to the retained components. They descend to by faithful flatness of Laurent polynomial extension.

For (2), if is monomial, a monomial in the remaining variables belongs to exactly when it belongs to . Hence commutes with intersections of monomial ideals; as a surjective homomorphism it also commutes with ideal powers. Apply to . Under the stated hypothesis the resulting components form a minimal irreducible primary decomposition, so (2) identifies the right side with .

Part (2) is deliberately conditional. Setting variables equal to zero is not asserted to preserve the Simis property without checking the image decomposition. In both applications below that check is explicit.

We use the following classical local criterion in its ideal case: a generically complete intersection ideal of grade at least two in a Cohen–Macaulay local ring is a complete intersection if all its ordinary-power quotients are Cohen–Macaulay. This is the rank-one specialization of Branco Correia–Zarzuela [5], Theorem 6.4. Its mechanism is Burch’s inequality, which forces equimultiplicity, followed by the fact that a generically complete intersection equimultiple ideal is a complete intersection. The latter criterion and its proof also appear in [6], Theorem 5.3.

Lemma 3.

Let . Suppose is Simis and its minimal primary components are irreducible of height . Then is a monomial complete intersection, has at most two minimal primes, and the exponents of any variable agree in all components containing it.

Proof.

Localize at the homogeneous maximal ideal . For every , the equality of symbolic and ordinary powers expresses as an intersection of powers of its primary components. Those powers are primary with height- radicals. There are no embedded associated primes, so the one-dimensional local quotient by has depth one and is Cohen–Macaulay. At every minimal prime, is generated by the pure powers in its corresponding irreducible component. It is therefore generically a complete intersection. The local ring is regular and has grade , so the preceding classical criterion applies.

The number of minimal monomial generators is unchanged on localizing at , as is seen from . Thus has exactly minimal monomial generators. Their supports are pairwise disjoint. In fact, if two shared a variable, that variable together with one variable from each of the other generators would give a prime of height at most containing , a contradiction. These disjoint nonempty supports occupy at most variables. All therefore have size one, with at most one exception of size two. The minimal primes select one variable from each generator support, giving at most two choices.

At such a prime, all unselected variables are units. The localized ideal is generated by the powers of the selected variables occurring in their unique monomial generators. Contraction gives the unique isolated primary component. In particular the exponent of a selected variable is independent of the prime in which it is selected.

Corollary 4.

An intersection of three irreducible primary ideals with radicals , , and in is not Simis.

Proof.

Lemma 3 with permits at most two minimal primes.

3. An obstruction on five variables

We give a uniform obstruction needed for the last support configuration. For a monomial ideal let be its Newton polyhedron, the convex hull of its exponent set. Equivalently it is the convex hull of its minimal generator exponents plus . For a decomposition into irreducible components, define

These irreducible polyhedra and their relation to integral closures are studied in Grisalde–Seceleanu–Villarreal [7], Section 7. Only the following necessary condition is required here.

Lemma 5.

Under the decomposition assumptions of Lemma 2, if is Simis, then . Every vertex of is then the exponent vector of a monomial in .

Proof.

The inclusion follows from . Take a rational and a positive integer such that every is integral for every relevant pair , and every is integral. Formula (3) gives . Since , we obtain . Rational points are dense in the rational polyhedron , and is closed, which proves equality.

Finally, every vertex of the convex hull of finitely many generator exponents plus the nonnegative orthant is one of those exponents. Indeed a nonzero orthant displacement places the point strictly between its undisplaced point and twice that displacement, while a nontrivial convex combination contradicts extremality.

Proposition 6.

For arbitrary positive integers , the ideal in with components

is not Simis.

Proof.

The four radicals are distinct incomparable triples. Use coordinates and put . The irreducible polyhedron has nonnegative coordinates and inequalities

If , take

This is feasible. The four equalities in (4) together with uniquely determine : the last three give , the first gives , and then . Hence is a vertex. Each of is strictly between zero and one, since its numerator is a positive summand of and the other two summands are positive.

If , instead take

Here , so the point is feasible. The first, third, and fourth equalities in (4), together with , uniquely determine it. It is again a vertex. Since and , we have , , and .

In both cases the vertex has all three first coordinates below the respective pure-power thresholds of . It is outside the union of the exponent orthants of , hence cannot be the exponent vector of a monomial in . Lemma 5 excludes the Simis property.

4. Proof of the height-three theorem

Assume is Simis. If two component supports intersect in two variables, their union has size four. Retain the components contained in that union by Lemma 2(1), and apply Lemma 3 with . Their weights agree on both common variables. We will use this conclusion repeatedly.

Suppose some variable nevertheless has different exponents in two components. Those supports must intersect in exactly that variable. After relabeling, and localizing to their union, write

with unequal -exponents and . Here denote variables; exponent letters in Proposition 6 will only be used after a separate relabeling. Every additional support containing would have the form with and . Its two-variable overlaps with and would force its -exponent to equal both and . Thus and are the only supports containing .

Call the other supports extras. They are triples in . There are at most two: if there are any, invert and apply Lemma 3 to the retained height-three components in four variables. Up to exchanging the pairs and permuting within each pair, there are four possibilities.

No extras

Exchange if necessary so that , and let be the -exponent in . The monomial

lies in . In the subring we have

A factor of a monomial involving only also involves only , so ordinary powers commute with this monomial contraction. The three generator products in degree two have exponent pairs , , and . None divides , because

Therefore , a contradiction.

One extra

By symmetry the extra has support . Set . The three image components have supports , , and , respectively. Their radicals are distinct and incomparable. Lemma 2(2) would make their intersection Simis, contrary to Corollary 4.

Two extras omitting variables from different pairs

The extras can be written and . They share the two variables , so the four-variable conclusion at the start of the proof makes their exponents on equal. Set . The image supports are

The last two ideals are identical, including their exponents. After deleting one duplicate, Lemma 2(2) again gives a Simis triangle, contradicting Corollary 4.

Two extras omitting variables from the same pair

The extras can be written and . Together with , they all contain . Pairwise four-variable localization makes the exponents on agree across all three. After relabeling the variables as , the four components have exactly the form of Proposition 6. That proposition supplies the final contradiction.

These possibilities exhaust the subsets of the four available extra triples of cardinality at most two. Thus every shared-variable exponent agrees. Let be this common exponent, choosing for any variable occurring in no component. Weighting commutes with monomial intersections, by the least-common-multiple formula for their generators. Hence

Corollary 5.5(b) of [8] now makes Simis. Conversely the same established weighting invariance proves that every standard weighting of a squarefree Simis ideal is Simis. Uniqueness of isolated primary components identifies the component exponents of such a weighting, so the three conditions of Theorem 1 are equivalent.0◻

5. Scope and further questions

The theorem removes the nonsquarefree weighting question in the entire unmixed height-three class under the irreducibility hypothesis. It leaves the separate squarefree Simis classification intact. It also does not assert that equality of just the second powers forces the conclusion: the dimension-one criterion uses all powers.

There is a general localization principle behind the argument. If all component heights are at most , a discrepancy on a shared variable is retained by localizing to the union of the two relevant supports, which has at most variables. Thus a counterexample to weighting rigidity in bounded height would already occur in that many variables. In unmixed height three the four-variable complete-intersection lemma makes the remaining support list short. For mixed heights at most three, or unmixed height four, additional configurations remain to be controlled. They are natural next cases; the present proof does not settle them.

The accompanying Python script checks the finite support reduction on five labeled variables and the two rational-vertex formulas using exact arithmetic. These calculations are supplementary: the proof above applies to all positive exponents without a finite cutoff. The manuscript and checks were prepared with OpenAI Codex. The originating system performed the internal audit; no independent human verification or peer review is represented.

References

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[2] Bijender and A. Kumar, Waldschmidt constant of monomial ideals and Simis ideals, preprint (2025), arXiv:2512.22940v1.

[3] P. Bordoloi, K. K. Das, and R. Kumar, Support-2 monomial ideals that are Simis, Journal of Algebraic Combinatorics 63 (2026), no. 3, Paper No. 40. Preprint version: arXiv:2504.07045v2.

[4] P. Bordoloi, K. K. Das, and R. Kumar, The Mendez–Pinto–Villarreal conjecture for some classes of monomial ideals, preprint (2026), arXiv:2607.25683v1.

[5] A. L. Branco Correia and S. Zarzuela, Some asymptotic properties of the Rees powers of a module, preprint (2004), arXiv:math/0408351v1.

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