Generic Regular Sequences

Resultants can be defined starting from a graded polynomial algebra Q ( T ) = Q [ T 0 , … , T n ] , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429064265/27e88de8-b4c2-47de-a1e8-3cf9bc1ec436/content/unequ8_60_1.tif"/> where T ₀,…,T_n are indeterminates of positive degrees γ₀,…,γ _n which are called weights, and generic homogeneous polynomials F ₀,…,F_n of positive degrees δ ₀,…,δ_n . That is, the F_j are polynomials F j = ∑ v ∈ N j U j v T v https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429064265/27e88de8-b4c2-47de-a1e8-3cf9bc1ec436/content/unequ8_60_2.tif"/> where N_j is a finite set of tuples ν = (v ₀,…,v_n ) ∊ ℕ ^{n +1}\{0} such that 〈 v , γ 〉       ≔       v 0 γ 0 + ⋯ + v n γ n = δ j https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429064265/27e88de8-b4c2-47de-a1e8-3cf9bc1ec436/content/unequ8_60_3.tif"/> and where U_jν are independent indeterminates over ℤ: the ground ring is the polynomial ring Q = ℤ [ U j v ] j , v . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429064265/27e88de8-b4c2-47de-a1e8-3cf9bc1ec436/content/unequ8_60_4.tif"/> .