Exercise 10.18

Let g 1 , g 2 , … , g m ∈ 𝔽 q [ x 1 , x 2 , … , x n ] be homogeneous polynomials of degree d and assume that n > md . Prove that there is nontrivial common zero. [Hint: Let f be as in Exercise 17 and consider the polynomial f ( g 1 ( x 1 , … , x n ) , … , g m ( x 1 , … , x m ) ) .]

Answers

Proof. Consider the polynomial h = f ( g 1 ( x 1 , … , x n ) , … , g m ( x 1 , … , x m ) ) ∈ 𝔽 q [ x 1 , … , x m ] . Then h is homogeneous of degree md . Since n > md , the Chevalley’s Theorem (Corollary of Theorem 1) shows that there is a non trivial zero a ¯ = ( α 1 , … , α m ) ∈ 𝔽 q m ∖ { ( 0 , … , 0 ) } of h , so that

f ( g 1 ( α 1 , … , α n ) , … , g m ( α 1 , … , α m ) ) = 0 , ( α 1 , … , α m ) ∈ 𝔽 q m ∖ { ( 0 , … , 0 ) } .

Then

f ( β 1 , … , β m ) = 0 , where  β i = g 1 ( α 1 , … , α n ) ∈ 𝔽 q .

Since f has no trivial zero by Exercise 17, we obtain β 1 = ⋯ = β m = 0 , that is

g 1 ( α 1 , … , α n ) = … = g m ( α 1 , … , α m ) = 0 , ( α 1 , … , α m ) ∈ 𝔽 q m ∖ { ( 0 , … , 0 ) } .

This proves that here is nontrivial common zero for g 1 , … g m , if n > md . □

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2022-07-19 00:00
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