Let fx y be a polynomial 1 Suppose every line through the or

Let f(x, y) be a polynomial.

(1) Suppose every line through the origin intersects f with multiplicity at least 2 there. Prove that every monomial appearing in f has degree at least 2.

(2) Suppose every line through the origin intersects f with multiplicity at least 3 there. Prove that every monomial appearing in f has degree at least 3.

(3) Suppose every line through the origin intersects f with multiplicity at least d there. Prove that every monomial appearing in f has degree at least d.

Solution

(3) obviously implies (1) and (2). Suffices to prove (3)

The intersection multiplicity is the order of 0 (at the intersection of any line through the origin and the curve f) is the multiplicity of the root x=0 of the polynomial f(x,mx).

As the intersection multiplicity is at least d , f(x,mx) as a polynomial in x is divisible by at least x^d . As the polynomial is homogeneous (it passes through the origin) , [f(x,mx) = x^d (g(m,x))] each monomial must be divisible by x^d .Hence the result.