Symbolic-Ordinary Discrepancy Module

• The discrepancy module formalizes and quantifies the gap between symbolic and ordinary powers of ideals by comparing I^(m) and I^m in commutative algebra.
• It employs combinatorial techniques and topological criteria to analyze filtration equivalence, stabilization of Rees algebras, and shifts in syzygies.
• It bridges abstract algebra with computational semantics by mapping symbolic descriptions to numerical features in object identification tasks.

The symbolic-ordinary discrepancy module formalizes, quantifies, and operationalizes the difference between symbolic powers and ordinary powers of ideals acting on modules, as well as the gap between symbolic abstractions and raw measurements in computational systems. In commutative algebra, the symbolic-ordinary discrepancy is measured by examining the containment $I^{(m)} \subseteq I^m$ for powers of an ideal $I \subset R$. The structure and properties of this module impact Rees algebras, regularity, syzygies, and the topological equivalence of induced filtrations, with analogous roles in computational perception for resolving differences between symbolic object descriptions and their numerical features.

1. Definitions: Discrepancy Modules and Filtrations

The symbolic-ordinary discrepancy is classically measured via the module

$M = \bigoplus_{m \geq 0} \frac{I^{(m)}}{I^m}$

where $I^{(m)}$ is the $m$th symbolic power, typically defined via intersections of powers of the minimal primes of $I$, and $I^m$ is the $m$th ordinary power, generated by all products of $m$ elements of $I$ (Keane et al., 2015, Nagel et al., 2015). Each graded piece $M_m$ captures the deviation in containment at level $m$; nonzero $M_m$ signifies $I^{(m)} \not\subseteq I^m$.

In the context of modules, given a Noetherian ring $R$, ideal $I$, and finitely generated $R$-module $N$, the $I$-adic and $I$-symbolic filtrations of $N$ define respective topologies:

• $I$–adic: $\{ I^n N \}_{n\geq1}$
• $I$–symbolic: $\{ (IN)^{(n)} \}_{n\geq1}$

These two topologies are equivalent iff for every $n$ there exists $m$ such that $(IN)^{(m)} \subseteq I^nN$, which is equivalent to the existence of finitely many nonzero components in the corresponding discrepancy module (Azari et al., 2016).

In computational semantics, a discrepancy module is instantiated as the statistical module mapping between symbolic object predicates (such as "red," "left," or "tall") and raw perceptual features. The module operationalizes the mapping by learning grounding functions and combining predicate evidence multiplicatively in identification pipelines (Baisero et al., 2017).

2. Structural Properties: Containment, Vanishing, and Generators

The symbolic-ordinary discrepancy is governed by explicit containment results. For axis-union ideals $I_{2,n} = (x_ix_j \mid 1 \leq i < j \leq n)$:

$$I_{2,n}^{(r_m)} \subseteq I_{2,n}^m, \quad r_m = \lceil (2 - 2/n)m \rceil$$

holds for all $m\geq1$, with $M_r = 0$ for $r \geq r_m$; thus, the gap module vanishes above a critical exponent (Keane et al., 2015). For Fermat ideals, containment fails generically ($I^{(3)} \not\subseteq I^2$ for $n \geq 3$), so nonvanishing of $M_3$ reflects genuine symbolic-ordinary discrepancy (Nagel et al., 2015).

Generators of $M$ correspond to minimal generators of symbolic powers $I^{(r)}$ that do not lie in $I^r$. These have explicit combinatorial and homological descriptions via Hilbert–Burch matrices and specialization to monomial ideals. The Noetherian property of symbolic Rees algebras implies $M$ is finitely generated (Nagel et al., 2015).

3. Topological Equivalence and Hartshorne-type Criteria

The interplay between symbolic and ordinary powers is encoded via topological equivalence of the induced filtrations. The foundational criteria are:

• The $I$-adic and $I$-symbolic topologies on $N$ are equivalent iff for all minimal associated primes $\mathfrak{p} \in \mathrm{mAss}_R(N/IN)$, the $\mathfrak{p}$-adic and $\mathfrak{p}$-symbolic topologies are equivalent (Azari et al., 2016).
• For a single height-one prime $\mathfrak{p}$, equivalence holds iff for each $z \in \mathrm{Ass}_{\widehat{R}}(\widehat{N})$, there exists $q \in \mathrm{Supp}(\widehat{N})$ such that $z \subseteq q$ and $q \cap R = \mathfrak{p}$.
• If all $\mathfrak{p}$-adic and $\mathfrak{p}$-symbolic topologies coincide for height-one $\mathfrak{p}$ in $\mathrm{Supp}(N)$, then $N$ is unmixed and $\mathrm{Ass}_R(N)$ is unique.

Modules with locally unmixed properties and ideals generated by the correct number of elements show automatic symbolic-ordinary equivalence, further collapsing the discrepancy module.

4. Homological and Algebraic Implications

The discrepancy module $M$ controls secondary invariants such as Castelnuovo–Mumford regularity, Hilbert functions, and syzygies. In Fermat configurations, regularity differences $\reg(I^{(r)}) - \reg(I^{r})$ are linear in $r$ for $r \gg 0$ and correlate with new syzygies in $M_r$. Minimal free resolutions are explicitly described for both ordinary and symbolic powers:

• Ordinary: Multistep resolutions extracted from almost complete intersection structure.
• Symbolic: Two-step Hilbert-Burch type resolutions for symbolic powers indexed by $kn+j$ (Nagel et al., 2015).

For containment relations, degrees where $M_r$ vanishes signal stabilization of symbolic powers within ordinary powers and yield finite presentations for symbolic Rees algebras.

5. Discrepancy Modules in Symbolic-Numeric Bridging

Outside pure algebra, the concept generalizes to the “symbolic-ordinary module” bridging high-level symbolic descriptions and raw sensory data in intelligent systems. In object identification, the discrepancy module operationalizes grounding of predicates (e.g., "red," "left," "tall") to perceptual features (e.g., hue, centroid, height):

• Feature grounding functions $\phi_s(o; \env)$ encode the mapping $s \mapsto$ measurement for each object $o$ in environment $\env$ (Baisero et al., 2017).
• Aggregation via log-linear models enables ensemble learning, leveraging weak or generic predicates.
• Learning proceeds by minimizing average KL divergence between the symbolic posterior and the perceptual posterior, regularized via feature partitioning.

Evaluation on the PR2 robotic platform achieved ~93% successful symbolic identifications and grasping in arbitrary block arrangements, demonstrating practical resolution of symbolic-ordinary mismatch in real-world perception.

6. Global Criteria and Reduction to Minimal Primes

The hierarchy of criteria for vanishing symbolic-ordinary discrepancy reflects a reduction from local to global tests:

• Testing equivalence for minimal associated primes $\mathrm{mAss}_R(N/IN)$.
• Invoking asymptotic primes $A^*(I,N)$, defined as the stable set of associated primes of $N / I^nN$ for $n \gg 0$, with the criterion that if the $\mathfrak{p}$-adic completion of $N_\mathfrak{p}$ is associated to a single prime for each $\mathfrak{p} \in A^*(I,N)$, the topologies coincide (Azari et al., 2016).

Each step reflects elimination of embedded or extraneous primes in the symbolic powers, with the global reduction showing that discrepancy is minimized when ideals and modules are unmixed and primary decompositions align.

7. Examples and Combinatorial Containment Proofs

Elementary proofs of containment relations and discrepancy module vanishing for monomial ideals rely solely on primary decomposition and inequality conditions on exponents:

• For $I_{2,n}$, a monomial $x^a$ belongs to $I_{2,n}^{(m)}$ iff $\sum_{j \neq i} a_j \geq m$ for all $i$, and to $I_{2,n}^m$ iff, additionally, $\sum_j a_j \geq 2m$.
• Aggregating $n$ inequalities yields the critical exponent $r_m$ signaling vanishing of the discrepancy module.
• Examples for $n=3,4$ illustrate sharp degrees where symbolic powers enter ordinary powers (Keane et al., 2015).

These combinatorial methods foreground a minimal set of facts for identifying, quantifying, and controlling symbolic-ordinary discrepancy across algebraic and computational paradigms.