Difference between revisions of "Polynomial"
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== Definition == | == Definition == | ||
| − | Let <math>n</math> and <math>d</math> be positive integers. Any polynomial of degree <math>d</math> on <math>n</math> over the field <math>\mathbb{F}_2</math> can be viewed as a Boolean function by mapping the identity of <math>\mathbb{F}_2</math> to 1 and the other element to -1. | + | Let <math>n</math> and <math>d</math> be positive integers. Any polynomial of degree <math>d</math> on <math>n</math> over the field <math>\mathbb{F}_2</math> can be viewed as a Boolean function by mapping the identity of <math>\mathbb{F}_2</math> to 1 and the other element to -1. |
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| + | Note: The polynomial function is different from the [[Fourier representation]]. While every polynomial over <math>\mathbb{F}_2</math> can be treated as a Boolean function, in the Fourier decomposition the polynomials are taken over <math>\mathbb{R}</math>; the coefficients are therefore real, and need to be chosen carefully so that the image lies in <math>\{-1,1\}</math>. Not every real multivariate polynomial is the Fourier decomposition of a Boolean function. | ||
== Properties == | == Properties == | ||
Latest revision as of 07:40, 1 October 2018
Definition
Let [math]n[/math] and [math]d[/math] be positive integers. Any polynomial of degree [math]d[/math] on [math]n[/math] over the field [math]\mathbb{F}_2[/math] can be viewed as a Boolean function by mapping the identity of [math]\mathbb{F}_2[/math] to 1 and the other element to -1.
Note: The polynomial function is different from the Fourier representation. While every polynomial over [math]\mathbb{F}_2[/math] can be treated as a Boolean function, in the Fourier decomposition the polynomials are taken over [math]\mathbb{R}[/math]; the coefficients are therefore real, and need to be chosen carefully so that the image lies in [math]\{-1,1\}[/math]. Not every real multivariate polynomial is the Fourier decomposition of a Boolean function.
Properties
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