Theorems · Theorem · commutative algebra
norm_root_le_spectralValue
∀ {K : Type u_2} [inst : NormedField K] {L : Type u_3} [inst_1 : Field L] [inst_2 : Algebra K L] {f : AlgebraNorm K L},
IsPowMul ⇑f →
IsNonarchimedean ⇑f → ∀ {p : Polynomial K}, p.Monic → ∀ {x : L}, (Polynomial.aeval x) p = 0 → f x ≤ spectralValue pThe norm of any root of p is bounded by the spectral value of p. See
[S. Bosch, U. Güntzer, R. Remmert, Non-Archimedean Analysis (Proposition 3.1.2/1(1))]
[bosch-guntzer-remmert].
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 199 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- NormedFieldFieldAlgebra
Around this declaration
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Cites66
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
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- Realstatement and proof · cited by 25,697
- Algebrastatement and proof · cited by 11,388
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- Polynomialstatement and proof · cited by 5,681
- Norm.normproof · cited by 5,413
- Finset.sumproof · cited by 5,195
- Set.rangeproof · cited by 4,705
- AlgHomstatement · cited by 3,236
- mul_commproof · cited by 2,262
- map_zeroproof · cited by 1,614
- add_commproof · cited by 1,535
Cited by3
Results whose statement or proof uses this declaration.
- spectralNorm_eq_iSup_of_finiteDimensional_normalproof · cited by 2
- max_norm_root_eq_spectralValueproof · cited by 1
- norm_le_spectralNormproof · cited by 0