isWeightedHomogeneous_zero_iff_weightedTotalDegree_eq_zero

English

Compatibility of weighted homogeneous components with DirectSum decomposition, expressing coordinatewise behavior of the component map.

Русский

Совместимость компонент взвешенной однородности с разложением по DirectSum, выражающая по-координатно поведение отображения компоненты.

LaTeX

$$$weightedHomogeneousComponent\, w\, m\, (DirectSum.coeLinearMap\, (weightedSubmodule)) = (DFinsupp.map)\; ?$$$

Lean4

/-- A multivariate polynomial is weighted homogeneous of weighted degree zero if and only if
  its weighted total degree is equal to zero. -/
theorem isWeightedHomogeneous_zero_iff_weightedTotalDegree_eq_zero {p : MvPolynomial σ R} :
    IsWeightedHomogeneous w p 0 ↔ p.weightedTotalDegree w = 0 :=
  by
  rw [weightedTotalDegree, ← bot_eq_zero, Finset.sup_eq_bot_iff, bot_eq_zero, IsWeightedHomogeneous]
  apply forall_congr'
  intro m
  rw [mem_support_iff]

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