Definition 25

A semi-section of the semi-bundle ℒ = ( E , M , F , π ) ℒ 𝐸 𝑀 𝐹 𝜋 {\cal L}=\left(E,M,F,\pi\right) caligraphic_L = ( italic_E , italic_M , italic_F , italic_π ) is defined by

π ∘ s ∘ π = π . 𝜋 𝑠 𝜋 𝜋 \pi\circ s\circ\pi=\pi. italic_π ∘ italic_s ∘ italic_π = italic_π . (49)

A reflexive semi-section satisfies to the additional condition

s ∘ π ∘ s = s . 𝑠 𝜋 𝑠 𝑠 s\circ\pi\circ s=s. italic_s ∘ italic_π ∘ italic_s = italic_s . (50)
Definition 1

Let ℱ ℱ {\cal F} caligraphic_F be a set of polynomials of k ⁢ [ X 1 , … , X n ] 𝑘 subscript 𝑋 1 normal-… subscript 𝑋 𝑛 k[X_{1},\dots,X_{n}] italic_k [ italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ] such that 0 0 belongs to ℱ ℱ {\cal F} caligraphic_F . Let 𝒬 𝒬 {\cal Q} caligraphic_Q be a subset of k n superscript 𝑘 𝑛 k^{n} italic_k start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT . 𝒬 𝒬 {\cal Q} caligraphic_Q is called a correct test sequence (or questor set) for ℱ ℱ {\cal F} caligraphic_F if for any P ∈ ℱ 𝑃 ℱ P\in{\cal F} italic_P ∈ caligraphic_F the following implication holds :

P ⁢ ( x ) = 0 ⁢ f ⁢ o ⁢ r ⁢ a ⁢ l ⁢ l ⁢ x ∈ 𝒬 ⟹ P = 0 . 𝑃 𝑥 0 𝑓 𝑜 𝑟 𝑎 𝑙 𝑙 𝑥 𝒬 ⟹ 𝑃 0 P(x)=0\;for\;all\;x\in{\cal Q}\Longrightarrow P=0\ . italic_P ( italic_x ) = 0 italic_f italic_o italic_r italic_a italic_l italic_l italic_x ∈ caligraphic_Q ⟹ italic_P = 0 .