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Homomorphism polynomials from symmetric circuits

Lax461425.PatternsFromCircuits · concepts/Lax461425/PatternsFromCircuits.lean · lax-461425

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    Natural Language Statement

    Theorem

    Let Δ≥1\Delta \geq 1, k≥2k \geq 2, and n≥2k+1n \geq 2k + 1. If a polynomial p∈K[Xn]p \in \mathbb{K}[\mathcal{X}_n] is computed by a reduced matrix-symmetric circuit of support size ≤k\leq k and support depth ≤Δ\leq \Delta, then pp is a linear combination of homomorphism polynomials hom⁡F,n\hom_{F,n} for patterns FF with fwΔ(F)≤k−1\mathrm{fw}_\Delta(F) \leq k - 1.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
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    1import Mathlib.LinearAlgebra.Span.Defs
    2import Lax461425.Supports
    3import Lax461425.Forestwidth
    4
    5/-!
    6---
    7title: Homomorphism polynomials from symmetric circuits
    8type: theorem
    9---
    10Let Δ≥1\Delta \geq 1, k≥2k \geq 2, and n≥2k+1n \geq 2k + 1. If a polynomial
    11p∈K[Xn]p \in \mathbb{K}[\mathcal{X}_n] is computed by a reduced matrix-symmetric circuit of support
    12size ≤k\leq k and support depth ≤Δ\leq \Delta, then pp is a linear combination of homomorphism
    13polynomials hom⁡F,n\hom_{F,n} for patterns FF with fwΔ(F)≤k−1\mathrm{fw}_\Delta(F) \leq k - 1.
    14-/
    15
    16namespace Lax461425.PatternsFromCircuits
    17
    18open Circuits MatrixSymmetry Supports HomomorphismPolynomials Forestwidth
    19
    20/-- A polynomial computed by a reduced matrix-symmetric circuit of support size `≤ k` and
    21support depth `≤ Δ` is a linear combination of homomorphism polynomials of patterns of
    22`Δ`-forestwidth at most `k - 1`. -/
    23axiom mem_span_hom {K : Type*} [Field K] [CharZero K] {Δ k n : ℕ} (hΔ : 1 ≤ Δ) (hk : 2 ≤ k)
    24 (hn : 2 * k + 1 ≤ n) (C : Circuit (Fin n × Fin n) K)
    25 (hred : C.IsReduced) (hsym : IsMatrixSymmetric C) (hsize : supportSize C ≤ k)
    26 (hdepth : supportDepth C ≤ Δ) {p : MvPolynomial (Fin n × Fin n) K} (hp : C.Computes p) :
    27 p ∈ Submodule.span K {q | ∃ F : Pattern, ForestwidthLE Δ (k - 1) F ∧ hom K F n = q}
    28
    29end Lax461425.PatternsFromCircuits
    30
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