Projective Hyperquadrics

Let \( F(X_0,X_1,\ldots ,X_n)=\sum _{i_h\geq 0,\;i_0+\cdots +i_n=d} a_{i_0i_1\ldots i_n}X_0^{i_0}X_1^{i_1}\cdots X_n^{i_n}\in { }_{\mathrm {h}}K[X_0,X_1,\dotsc , X_n]_d^*.\) As observed in Example 1.33 -(2), \({ }_{\mathrm {h}}K[X_0,X_1,\dotsc , X_n]_d\) is a K-vector space.