Waldschmidt Constant of Monomial Ideals

• The Waldschmidt constant is an asymptotic invariant that measures the growth rate of least degrees in symbolic powers of monomial ideals, characterized via linear programming.
• It connects algebra with combinatorial optimization and convex geometry, enabling explicit computations for squarefree ideals and providing bounds using Newton polyhedra.
• The study offers sharp lower and upper bounds while linking fractional chromatic numbers to symbolic power containment, thus enhancing practical computational methods.

The Waldschmidt constant of a monomial ideal is a fundamental asymptotic invariant capturing the growth rate of the least degrees of forms in symbolic powers. Its study connects combinatorial optimization, convex geometry, and the fine structure of symbolic versus ordinary powers within the field of commutative algebra and algebraic geometry. For monomial ideals, especially squarefree ones, the Waldschmidt constant admits an explicit description via linear programming, which also links it to concepts from fractional graph theory and polyhedral geometry. The following article collects rigorous definitions, computational methods, bounds, and prominent examples arising from contemporary research literature.

1. Formal Definition and Existence

Given a standard $\mathbb{N}$-graded polynomial ring $R = K[x_1, \dots, x_n]$ and a (homogeneous, typically monomial) ideal $I \subset R$, the $m$-th symbolic power of $I$ is defined as

$$I^{(m)} = \bigcap_{P \in \operatorname{Ass}(I)} \bigl(I^m R_P \cap R\bigr),$$

where $\operatorname{Ass}(I)$ is the set of associated primes of $I$. The initial degree $\alpha(J)$ of a homogeneous ideal $J$ is the smallest $d$ such that $J$ contains a nonzero element of degree $d$.

The Waldschmidt constant of $I$ is the asymptotic slope

$$\widehat{\alpha}(I) = \lim_{m \to \infty} \frac{\alpha\bigl(I^{(m)}\bigr)}{m} = \inf_{m \geq 1} \frac{\alpha\bigl(I^{(m)}\bigr)}{m}.$$

The existence of this limit follows from the subadditivity of $\alpha(I^{(m)})$ in $m$ (Drabkin et al., 2018, Bijender et al., 28 Dec 2025).

This definition extends to any Noetherian graded filtration of homogeneous ideals, where the limit persists and reflects the asymptotic behavior of minimum degrees in the filtration (Kumar et al., 2024).

2. Linear Programming and Polyhedral Characterization

For squarefree monomial ideals $I$ with minimal primary decomposition $I = P_1 \cap \dots \cap P_s$, where each $P_j$ is a monomial prime, the Waldschmidt constant is succinctly computed via linear programming (Bocci et al., 2015, Camarneiro et al., 2021, Grisalde et al., 2021): $$\widehat{\alpha}(I) = \min \left\{ y_1 + \cdots + y_n \mid A y \geq 1,\, y \geq 0 \right\}$$ where $A$ is an $s \times n$ 0–1 matrix with $A_{j,i} = 1$ iff $x_i \in P_j$.

For general monomial ideals, one associates an "asymptotic Newton polyhedron" constructed from the decomposition (primary or irreducible) of $I$. The minimization

$$\widehat{\alpha}(I) = \min\left\{ \sum_{i=1}^n y_i : (y_1, \dots, y_n) \in \mathcal{P}(I) \right\}$$

takes place over the corresponding convex body $\mathcal{P}(I)$, which, in the symbolic case, is an intersection of the Newton polyhedra of the combined symbolic primary components (Camarneiro et al., 2021, Grisalde et al., 2021). This geometric framework allows uniform treatment of both symbolic and integral closure filtrations.

The optimal value is always attained at a rational vertex of the polyhedron, providing an effective computational route for $\widehat\alpha(I)$ (Grisalde et al., 2021).

3. Connections to Graph Theory, Hypergraphs, and Fractional Chromatic Numbers

Given a squarefree monomial ideal $I$, one may interpret $I$ as the edge ideal of a hypergraph $H = (V,E)$, with vertices being variables and edges reflecting the supports of minimal generators. The Waldschmidt constant then admits an alternative description in terms of the fractional chromatic number $\chi_f(H)$: $$\widehat{\alpha}(I) = \frac{\chi_f(H)}{\chi_f(H) - 1}$$ for $H$ with at least one nontrivial edge (Bocci et al., 2015).

The relevant LP for $\chi_f(H)$ is: $$\min \left\{ \sum_j y_{W_j} : \sum_{W_j \ni x_i} y_{W_j} \geq 1 \; \forall i,\, y_{W_j} \geq 0 \right\}$$ where $W_j$ runs over independent sets in $H$. This duality between algebraic invariants and combinatorial optimization is central in recent research and leads to sharp values for numerous classes such as Stanley–Reisner ideals of uniform matroids (Bocci et al., 2015).

4. Rigorous Lower and Upper Bounds

A fundamental lower bound, verified for broad classes of monomial ideals, and proven in full generality for squarefree cases, is: $$\widehat{\alpha}(I) \geq \frac{\alpha(I) + h - 1}{h}$$ where $h$ is the maximal height of associated primes ("big-height") of $I$ (Bocci et al., 2015, Bijender et al., 28 Dec 2025). This generalizes Chudnovsky's conjectural lower bound for symbolic powers of points in projective space.

For monomial ideals admitting a standard linear weighting—that is, $I = J_w$, a weighted lift of a squarefree monomial ideal $J$—subadditivity and polarization arguments keep the lower bound sharp even outside the squarefree case (Bijender et al., 28 Dec 2025). In the special case of "whiskered hypergraphs" and certain Simis ideals, the bound is also achieved (Bijender et al., 28 Dec 2025).

Upper bounds for the Waldschmidt constant utilize the asymptotic Hilbert polynomial $aHP_I(t)$, defining auxiliary polynomials whose real roots control the constant: $Q_I(t) = \binom{t+n}{n} - aHP_I(t)$ with $\widehat{\alpha}(I) \leq$ largest real root of $Q_I^{(c)}(t) = 0$, where $c$ is dictated by the eventual depth of symbolic powers. This approach leverages the Hilbert polynomial asymptotics and applies to any radical monomial ideal with linearly bounded symbolic regularity (LBSR), a condition satisfied by all known monomial ideals (Dumnicki et al., 2015).

5. Special Families, Formulae, and Notable Examples

Several classes admit closed formulas for their Waldschmidt constants:

• Cover ideals of graphs: For a finite simple graph $G$ with cover ideal $J(G)$, the symbolic Rees algebra is generated in degree at most 2. Thus, $$\widehat\alpha(J(G)) = \min\{\alpha(J(G)), \alpha(J(G)^{(2)})/2\}$$. This produces, for a complete graph $K_r$, $\widehat\alpha(J(K_r)) = r/2$, and for an odd cycle $C_{2k+1}$, $\widehat\alpha(J(C_{2k+1})) = (2k+1)/2$ (Drabkin et al., 2018, Kumar et al., 2024).
• Principal squarefree Borel ideals: For $I = \operatorname{sfBorel}(m)$, sharp bounds (and in many cases, exact formulas) exist in terms of the support and combinatorics of Borel moves. Every rational $\geq 1$ is the Waldschmidt constant of some such ideal (Moreno et al., 2021).
• Stanley–Reisner ideal of the uniform matroid: $I_{n+1,c}$ generated by all squarefree monomials of degree $n+2-c$ satisfies $\widehat{\alpha}(I_{n+1,c}) = \frac{n+1}{c}$ (Bocci et al., 2015).
• Monomial curves: For the defining ideal $\mathfrak{p}$ of the monomial curve $C(2q+1, 2q+1+m, 2q+1+2m) \subset \mathbb{A}^3$, one obtains

$$\widehat\alpha(\mathfrak{p}) = \begin{cases} 2(2q+1+m) & \text{if } (q,m) \neq (1,1)\ 15/2 & \text{if } (q,m) = (1,1) \end{cases}$$

(D'Cruz, 2019).

6. Algorithmic and Computational Aspects

Given the LP characterization, $\widehat{\alpha}(I)$ is effectively computable for squarefree monomial ideals of moderate size. The number of constraints equals the number of associated primes, and the variables correspond to the variables of $R$. For general (possibly non-squarefree) monomial ideals, reduction via polarization and linear weightings allows extension of these techniques, with symbolic powers accessible via the corresponding Newton polyhedra (Camarneiro et al., 2021, Grisalde et al., 2021, Bijender et al., 28 Dec 2025).

For monomial ideals with Noetherian symbolic Rees algebra generated in degree $\leq n$, Drabkin–Guerrieri established that it suffices to compute the first $n$ symbolic powers: $$\widehat{\alpha}(I) = \min_{1 \leq m \leq n} \frac{\alpha(I^{(m)})}{m}$$ (Drabkin et al., 2018).

7. Broader Implications and Interconnections

The Waldschmidt constant interfaces with several major invariants:

• Resurgence: The ratio $\frac{\alpha(I)}{\widehat\alpha(I)}$ provides a lower bound for the resurgence $\rho(I)$, encoding containment thresholds for symbolic and ordinary powers (Bocci et al., 2015, Drabkin et al., 2018).
• Symbolic polyhedra and convex geometry: Convex-geometric language generalizes the study of symbolic blowups and provides a natural setting for understanding asymptotic invariants and their algorithmic computation (Camarneiro et al., 2021).
• Fractional invariants and graph theory: For edge ideals and cover ideals, the Waldschmidt constant translates directly to fractional covering and coloring parameters, establishing deep links between combinatorics and the algebraic geometry of monomial schemes (Bocci et al., 2015).

Recent developments confirm conjectured lower bounds for broad families—such as those with standard linear weighting or Simis structure—demonstrating the sharpness and robustness of the asymptotic approach in both commutative algebra and combinatorics (Bijender et al., 28 Dec 2025). Upper bounds, by contrast, utilize the asymptotic Hilbert polynomial, extending prior work and offering tight predictions in natural geometric settings (Dumnicki et al., 2015). The practical computability and direct combinatorial interpretation place the Waldschmidt constant as a central and explicitly accessible invariant among asymptotic measures of monomial ideals.