Citations:vanishing ideal

• 1985, Ernst Kunz, Introduction to Commutative Algebra and Algebraic Geometry, Springer Science & Business Media (Template:ISBN), page 33:

  Under the hypotheses made in 5.1, just as in the affine case for a nonempty projective K-variety $V\subset P^n(L)$ we define the vanishing ideal $\Im (V)\subset K[X_0,...,X_n]$ as the set of all polynomials that vanish at all points of V.

• 2013, Grigoriy Blekherman, Pablo A. Parrilo, Rekha R. Thomas, Semidefinite Optimization and Convex Algebraic Geometry, SIAM (Template:ISBN), page 465:

  Given a variety $W\subset k^n$, its vanishing ideal,
  $I(W):=\{f\in k[x]:f(p)=0\text{ for all }p\in W\}$
  is the set of all polynomials in $k[x]$ that vanish on W. Check that $I\subseteq I(V_k(I))$ and that $V_k(I(V_k(I)))=V_k(I)$.

• 2013, Wolfram Decker, Gerhard Pfister, A First Course in Computational Algebraic Geometry, Cambridge University Press (Template:ISBN), page 19:

  Our next step in relating algebraic sets to ideals is to define some kind of inverse to the map V:
  Definition 1.16 If $A\subset {𝔸}^n(K)$ is any subset, the ideal
  $I(A):=\{f\in K[x_1,...,x_n]|f(p)=0\text{ for all }p\in A\}$ is called the vanishing ideal of A.