Suppose there are k non-zero homogeneous polynomials P 1 </msup> ( . ) ,

Carly Cannon 2022-07-08 Answered
Suppose there are k non-zero homogeneous polynomials P 1 ( . ) , , P k ( . ), each of degree r in n variables, such that P j ( x 1 , , x n ) 0 for all ( x 1 , , x n ) 0, for all j [ k ]. Under what conditions (on the P j ( . )s) would there exist an α R + n such that P j ( α ) > 0 for all j [ k ]?
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Answers (2)

Zichetti4b
Answered 2022-07-09 Author has 13 answers
Let B be an open ball with center x 0 . Then B contains x 0 + i = 1 n t i e i for all i, where e i is the i-th unit vector and ε is a suitable positive number. Thus, if the polynomial P vanish on B, es geht that P ( u ) = 0 for all u B 1 × B 2 × × B n with all B i infinite.

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dream13rxs
Answered 2022-07-10 Author has 4 answers
A nonzero polynomial of degree r has some rth partial derivative which is a nonzero constant. That's impossible for a function which is zero on an open set.

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