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 ) &gt; 0 for all j ∈ [ k ]?

Question

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  )  &amp;gt;  0 for all   j  ∈  [  k  ]?

Zichetti4b · Accepted Answer

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.

dream13rxs · Accepted Answer

A nonzero polynomial of degree r has some rth partial derivative which is a nonzero constant. That&#039;s impossible for a function which is zero on an open set.

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

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