ABSTRACT
In this chapter, we concentrate on the existence and dimension of (possible) subspaces of P−1(0), where P is a K-valued polynomial on a Banach space X. It is clear that we must make restrictions in order that there be a chance of something interesting emerging. Namely, the polynomials P : K2 → K, P (x1, x2) = 1 + x21 + x22, Q : K2 → K, Q(x1, x2) = x31 + x32 and R : K2 → K, R(x1, x2) = x21 +x22 illustrate the three most basic requirements: • P shows that the polynomial in question must vanish at 0 ∈ X; • Comparing the polynomials P and R shows that the scalar field K = R
or C matters;
• Comparing the polynomials P and Q shows that the degree of the polynomial matters.
