My question was the following:Let E be a complement to the canonical k plane bundle is it true to say that E has no k dimensional sub bundle

thank you but this is not my question. My question was the following:

I think that the order power in Lojasiewicz type inequalities is sensitive in this question. So do you think that the Ulam stability of homogeneous polynomials in at least 3 variables can be solved with other approaches? however in 2 variable the subject is obvious because of factorization to polynomials with low(1 or 2) degree

Thank you. But i think that some thing is missing.Let the degree of polynomial is n. Then we need to know, on the unit spher, \delta is in the for O(\epslon^{n})

Thank you, But could you please explain why the last inequality hold?