Concept

# Characteristic function

Summary
In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
• The indicator function of a subset, that is the function ::\mathbf{1}_A\colon X \to {0, 1}, :which for a given subset A of X, has value 1 at points of A and 0 at points of X − A.
• There is an indicator function for affine varieties over a finite field: given a finite set of functions f_\alpha \in \mathbb{F}q[x_1,\ldots,x_n] let V = \left{ x \in \mathbb{F}q^n : f\alpha(x) = 0 \right} be their vanishing locus. Then, the function P(x) = \prod\left(1 - f\alpha(x)^{q-1}\right) acts as an indicator function for V. If x \in V then P(x) = 1, otherwise, for some f_\alpha, we have f_\alpha(x) \neq 0, which implies that f_\alpha(x)^{q-1} = 1, hence P(x) = 0.
• The characteristic function in convex analysis, closely
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