Citations:vanishing ideal


 * 1985, Ernst Kunz, Introduction to Commutative Algebra and Algebraic Geometry, Springer Science & Business Media (ISBN 9780817630652), page 33:
 * Under the hypotheses made in 5.1, just as in the affine case for a nonempty projective K-variety $$V \sub P^n(L)$$ we define the vanishing ideal $$\mathfrak{I}(V) \sub K[X_0, . . ., X_n]$$  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 (ISBN 9781611972283), page 465:
 * Given a variety $$W \sub k^n$$, its vanishing ideal, $$I(W) := \{f \in k[x] : f(p) = 0 \mbox{ for all } p \in W\}$$ is the set of all polynomials in $$k[x]$$ that vanish on W. Check that $$I \sube I(V_k(I))$$ 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 (ISBN 9781107612532), 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 $$A \sub \mathbb{A}^n(K)$$ is any subset, the ideal $$I(A) := \{f \in K[x_1,...,x_n]|f(p) = 0 \mbox{ for all } p \in A\}$$ is called the vanishing ideal of A.