integralNormalization_degree — Mathlib

English

The leadingCoeff smul of integralNormalization(p) equals scaleRoots(p, leadingCoeff(p)) as a generalization over a commutative semiring.

Русский

Смулинговая ведущий коэффициент интегральной нормализации равна scaleRoots(p, leadingCoeff(p)).

LaTeX

$$$\operatorname{leadingCoeff}(p) \cdot \operatorname{integralNormalization}(p) = \operatorname{scaleRoots}(p, \operatorname{leadingCoeff}(p))$$$

Lean4

theorem integralNormalization_degree : (integralNormalization p).degree = p.degree :=
  by
  apply le_antisymm
  · exact Finset.sup_mono p.support_integralNormalization_subset
  · rw [← degree_scaleRoots, ← integralNormalization_mul_C_leadingCoeff]
    exact (degree_mul_le _ _).trans (add_le_of_nonpos_right degree_C_le)

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