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9eeb9c7 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 | import Mathlib.Algebra.Polynomial.Degree.TrailingDegree
import Mathlib.Algebra.Polynomial.OfFn
import Mathlib.Basic.Complex.Basic
namespace OAI
namespace InvariantProjection
open Polynomial
theorem exists_nonzero_product_coeff_le
{R : Type*} [Semiring R] [NoZeroDivisors R]
(A B : R[X]) (k l : ℕ)
(hA : A.coeff k ≠ 0) (hB : B.coeff l ≠ 0) :
∃ n ≤ k + l, (A * B).coeff n ≠ 0 := by
have hA0 : A ≠ 0 := by
intro h
simp [h] at hA
have hB0 : B ≠ 0 := by
intro h
simp [h] at hB
refine ⟨(A * B).natTrailingDegree, ?_, ?_⟩
· rw [natTrailingDegree_mul hA0 hB0]
exact Nat.add_le_add
(natTrailingDegree_le_of_ne_zero hA)
(natTrailingDegree_le_of_ne_zero hB)
· exact coeff_natTrailingDegree_ne_zero.mpr (mul_ne_zero hA0 hB0)
theorem exists_nonzero_product_coeff_le_two_mul
(A B : ℂ[X]) (K : ℕ)
(hA : A.coeff K ≠ 0) (hB : B.coeff K ≠ 0) :
∃ n ≤ 2 * K, (A * B).coeff n ≠ 0 := by
simpa [two_mul] using exists_nonzero_product_coeff_le A B K K hA hB
end InvariantProjection
end OAI
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