Symbolic defect and normalization of whiskered cover ideals
Henry Zweiman
September 8, 2026
Abstract
Let be the cover ideal of the full whiskering of a graph on vertices. We express its symbolic defect and integral symbolic defect as differences between the Ehrhart functions of the fractional and ordinary stable-set polytopes and the Hilbert function of the stable-set semigroup. For every nonbipartite core graph the two defects are asymptotic, with common leading term and positive coefficient. Their difference is eventually polynomial of degree at most , without any normality hypothesis. This bound is sharp: when the complement of is a disjoint union of two odd cycles of lengths at least five, we determine the difference in every degree by a single binomial coefficient, and give a composition formula for any number of odd-cycle components. We also give closed formulas for the full symbolic defect of whiskered complete graphs.
1. Introduction and statements
For a homogeneous ideal in a polynomial ring, the symbolic defect measures the number of generators needed to account for the difference between symbolic and ordinary powers. The integral symbolic defect introduced by Oltsik [7, Definition 4.1] is
Here a bar denotes integral closure and the minimal number of module generators. These two counts need not agree. Oltsik asks about their growth comparison and the degrees and coefficients of their eventual quasipolynomials [7, Section 7]. We obtain an exact description for cover ideals of fully whiskered graphs, including graphs whose stable-set semigroup rings are not normal.
We write . Throughout, is a finite simple graph with specified vertex set , where . Isolated vertices are allowed. Its full whiskering has vertices , the core edges for , and the pendant edges . Over an arbitrary field , put
Write and .
Let be the independent sets of , including the empty set. We use the polytopes and finite sets
The upper coordinate bounds in include isolated vertices. Define
All volumes below are Euclidean -dimensional volumes, with unit cube volume one. Thus the Ehrhart leading coefficient is , rather than .
Theorem 1.1.
For every graph as above and every ,
The difference is nonnegative, eventually polynomial, and . It vanishes for every if and only if has the integer decomposition property.
Both defects are eventually quasipolynomial with period dividing two. If is bipartite, both vanish identically. If is nonbipartite, then
In particular both parity branches have degree and leading coefficient , and .
The next result gives an exact correction in a nonnormal family. For a graph , its complement is denoted by ; this notation is distinct from integral closure of ideals.
Theorem 1.2.
Suppose , where are odd, and put . Then
Consequently the exponent in Theorem 1.1 is sharp. The correction first occurs in degree and has leading coefficient .
For complete cores, the normalization correction is zero and the full defect has a direct closed form.
Theorem 1.3.
For one has for all . For ,
In (4), is interpreted as zero. For , the leading coefficient as a function of is . For , all defects are zero.
Relation to previous work
The ordinary generators of whiskered cover ideals, their independent-set parametrization, and their analytic spread occur in Drabkin–Guerrieri [3, Proposition 5.7 and Theorem 5.8]. The additional observation used here is that every symbolic power is also generated in a single degree, and that ordinary products and integral closure can be compared inside the same affine slice.
Eventual quasipolynomiality with period dividing two for graph cover ideals is already known [2, Corollary 4.3]. The conditional recursion in [2, Theorem 5.5] does not apply to all whiskered graphs: [2, Remark 5.3] exhibits the whiskered triangle as a failure of its indecomposability hypothesis. Theorem 1.3 includes that example and concerns whiskered complete graphs, whereas [2, Theorem 5.7] treats unwhiskered complete graphs.
The polyhedral facts about fractional stable-set polytopes used below are established [4]. Conditional comparisons with integral closure appear in [7, Proposition 4.5] and, more recently, [1, Theorem 4.1]; the latter paper also gives coefficient constancy results under filtration and module hypotheses [1, Theorem 4.2]. Our conclusions are the explicit volume difference, unconditional ratio one for this whiskered class, and exact correction (3). They do not resolve the unrestricted growth questions for monomial ideals.
Nonnormality in Theorem 1.2 is already implied by [6, Theorem 1]. Preservation of multiplicity under semigroup normalization is standard; see also [5, Proposition 3.1]. Moreover, [5, Proposition 4.2] computes a translated free-face hole monoid and its Hilbert-series correction for a different connected edge-ring family built around two triangles. Our exact correction is a specific stable-set and symbolic-defect calculation with that conceptual precedent. We give elementary proofs, including the exclusion of holes away from the relevant facet.
2. The common degree slice
Since is squarefree with minimal primes indexed by the edges,
Thus if and only if
Lemma 2.1.
The minimal monomial generators of are precisely
In particular, every such generator has degree .
Proof.
In a minimal generator, no can exceed , because replacing it by preserves every inequality in (6). Once , a strict inequality permits decreasing the positive exponent . Hence minimality requires for every . Conversely, these equalities prevent decreasing any positive exponent: its pendant inequality would fail. Substituting in the core inequalities gives , exactly the asserted lattice set.
At , the lattice vectors in Lemma 2.1 are precisely the independent-set indicators. A product of such generators is with , and every element of occurs. These products all have degree , so distinct products are identified only when their exponent vectors are equal.
Let be the homogeneous maximal ideal of . If a monomial ideal is generated in one degree and is a monomial ideal, the minimal generators of are exactly the degree- generators of absent from . Their nonzero classes form a basis modulo : the latter has no degree- part, and the monomials present in the quotient are linearly independent. Applying this to , gives .
For a monomial ideal, a monomial belongs to its integral closure precisely when its exponent lies in its Newton polyhedron. This criterion can also be expressed by requiring a positive power of the monomial to belong to the same power of the ideal; see [7, Proposition 2.13]. The Newton polyhedron of is the convex hull of its generating exponent vectors plus . Its intersection with the total-degree hyperplane is exactly the affine image of under
Indeed the convex hull of all sums of independent-set indicators is . All these exponent vectors already have total degree , and a nonzero nonnegative addition would increase that degree. It follows that
Each coordinate-prime power is integrally closed, as is their intersection. Thus is integrally closed and . Applying the same degree- quotient argument gives . This argument does not assume that the entire ideal is generated in degree .
3. Normalization and asymptotic comparison
We include the semigroup estimate to make clear that no normality assumption on the stable-set ring is needed.
Lemma 3.1.
Let be generated by for , and let . Then
The difference is eventually a polynomial.
Proof.
The group generated by is all of : it contains and , and their differences give . The degree- elements of are exactly .
There exists a conductor element with . Here is a direct construction. If generate , subtract the floors of the coefficients in a nonnegative real expression for . The remainder is a lattice point in the bounded set . Its finitely many possible values belong to , and . Fullness of the generated group allows us to write with . Taking ensures for every , as required.
Write . Translation by injects into for , and hence
The last assertion uses Ehrhart’s theorem: is an integral, full-dimensional polytope, containing , so is a polynomial of degree . Finally is the Hilbert function of the standard graded affine semigroup algebra , and is therefore eventually polynomial by the Hilbert–Serre theorem. Their difference is eventually polynomial as well.
Lemma 3.2.
Every vertex of has coordinates in .
Proof.
Let be a vertex. Consider the graph on coordinates with , retaining the edges on which . A tight edge cannot join a fractional coordinate to a coordinate equal to zero or one. If a connected component of this tight graph is bipartite, add on one side and subtract it on the other. For sufficiently small positive and negative , every nontight inequality and every coordinate bound remains satisfied. This contradicts extremality. Every component therefore has an odd cycle. The tight equations around that cycle force all its coordinates to be , and the same equations propagate this value throughout the component.
Proposition 2.2 and Lemma 3.1 give the identities and correction estimate. Since satisfies the edge inequalities, its lattice points are exactly the independent-set indicators. Thus for every is precisely the integer decomposition property of .
By Lemma 3.2 and Ehrhart’s theorem, is a quasipolynomial of period dividing two, with both branches having leading term . Also
The asserted eventual quasipolynomiality and asymptotic expansions follow.
Suppose first that is bipartite with parts . For , assign vertex the first colors from , and vertex the last colors. Adjacent vertices receive disjoint color sets because . Each color class is therefore independent, giving a representation of in . Isolated vertices can be placed in either part. Hence and both defects vanish.
If is not bipartite, choose an odd cycle with vertices. On its vertex set the coordinate sum of any point of is at most . The point belongs to and violates that inequality, so . Both polytopes are full-dimensional. Their strict containment gives : moving a point of slightly toward an interior point of produces a small open ball in disjoint from the closed set . Thus . It follows that both parity branches have degree exactly , and that eventually. Dividing the two asymptotic expansions proves the ratio limit.
4. An exact correction from two odd cycles
Let be an odd cycle and a nonnegative integral demand vector on its vertices. Put and define
Repeated use of an edge is allowed in this capacitated matching problem.
Lemma 4.1.
One has . A noninteger value of occurs exactly when the full saturation equations have a nonnegative, nonintegral solution. In that case the solution is unique, every edge weight is a positive halfinteger, is odd, and .
Proof.
Choose an optimal vertex of the bounded fractional matching polytope. If an edge coordinate is zero, its positive support is a disjoint union of paths. The vertex-edge incidence matrix of a bipartite graph is totally unimodular: changing signs on the rows of one bipartition gives an oriented incidence matrix, whose square subdeterminants are . Thus the resulting active system with integral right-hand side has an integral vertex.
If every edge coordinate is positive, extremality requires every vertex inequality to be tight; otherwise there are fewer active constraints than edge variables. The incidence matrix of an odd cycle is nonsingular, as its homogeneous equations alternate signs and the odd closing condition forces zero. Solving successively shows that its saturated solution is either integral or has every coordinate in . In the latter case all coordinates are at least , and is odd. Round consecutive edge coordinates alternately down, up, down, up, ending down. All vertex sums remain unchanged except the one between the first and last edges, where the sum decreases by one. The resulting integral matching has weight .
This proves the floor identity. Conversely, any feasible nonintegral saturated solution attains the universal upper bound , and its uniform halfintegrality on an odd number of edges makes odd.
Set . Because both cycles have length at least five, the independent sets of are exactly the empty set, singletons, and edges of . For , put , and let be the two matching numbers on component . Then
Indeed a choice of edge weights leaves nonnegative residual demands filled by singleton weights. The total coefficient mass is ; empty-set weights fill any remaining mass up to . This argument works over the nonnegative reals for (8) and over the nonnegative integers for (9).
If both fractional optima are integral, the two tests coincide. If just one is a halfinteger, rounding the fractional threshold up to the integer again gives the ordinary threshold. Thus a hole can occur only when both fractional optima are halfintegers. By Lemma 4.1, each component is then fully saturated by its unique positive halfintegral edge vector, and its demand total is odd. The fractional threshold is , an integer, whereas the ordinary threshold is . Consequently a hole occurs exactly when and both saturation solutions are positive halfintegral. This also excludes holes away from the face .
Write these edge weights as , with . The block incidence matrix of the two odd cycles is nonsingular, so edge weights determine demands injectively. Conversely every such edge vector gives integral nonnegative demands, odd component totals, and the hole condition at degree . We have obtained a bijection between degree- holes and
Stars and bars gives (3), by Theorem 1.1. Its first degree and leading coefficient follow immediately.
Remark 4.2.
Equivalently, the Hilbert-series correction for this family is
The correction consists of a translate of the free monoid on the edge vectors of . For , its first values are in degrees through .
The same threshold argument gives a composition formula with any number of odd-cycle components. The factors below are the Hilbert series of single-component stable-set rings, so the formula uses ordinary Hilbert data to recover the entire normalization correction of their graph join.
Proposition 4.3.
Suppose , with every odd. Let
Then
The right-hand side is a formal power series with nonnegative exponents. For it equals
Proof.
For a demand vector on component , let be its minimum ordinary coefficient mass, and mark the component if is a halfinteger. By Lemma 4.1, the minimum fractional mass is on an unmarked component and on a marked one. In particular the single-component polytope has the integer decomposition property: for an integer degree, its fractional and ordinary feasibility thresholds agree.
The series counting all component demands by their minimum ordinary mass is , since the Hilbert function counts demands whose minimum mass is at most the given degree. A marked component has unique edge weights and
Thus counts the marked demands by , and counts the unmarked demands. All coefficients are nonnegative and each coefficient is finite, because total demand is at most twice the minimum mass.
For a demand on all components, put and let be the set of marked components. Independent sets of lie within a single component of its complement, so the ordinary minimum mass is and the fractional minimum mass is . Its missing degrees are exactly , for . Multiplying the component demand series and recording those degrees gives (10). Each factor starts in degree at least three, so the displayed negative powers cannot create negative exponents. The specialization follows directly.
For , independent sets are empty or singletons. Hence
It remains to count integral with and for distinct . Put . If every coordinate is at most , there are choices. Otherwise exactly one coordinate exceeds . If that coordinate is , every other coordinate ranges independently from to ; their mutual pair constraints are automatic. The additional count is
Separating even and odd and subtracting proves (4)–(5). The same partition works for , where the coordinate bound is retained even though there are no edges. For the graph is bipartite, so the defect vanishes. Finally, for the power-sum estimate gives leading coefficient . It is positive because for .
In particular the whiskered triangle has and . The calculation includes the obstruction in [2, Remark 5.3] without using the indecomposability recursion.
Verification and scope
The proofs above are independent of computation. Ancillary scripts check direct minimality of symbolic exponent vectors on all core graphs with at most three vertices and powers at most three; ordinary and symbolic counting on all labeled graphs with at most five vertices in powers one through three; and the complete-core formulas for . A separate check compares fractional matching vertex enumeration with exact integer matching dynamic programming on demand vectors on , and verifies the two-component correction through degree seven. All checks pass. The fractional vertex calculation uses numerical linear algebra with a checked halfinteger tolerance; Lemma 4.1 supplies the exact argument.
The results concern the fully whiskered cover-ideal class. They neither classify normal stable-set polytopes nor determine the minimal quasiperiod for arbitrary cores. The complete-core formula and the nonnormal correction family are parts of the same comparison theorem.
References
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