1.3: Polinomios simétricos

Última actualización
Guardar como PDF

Page ID: 111012

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

Hasta ahora, hemos considerado objetos geométricos. Tengamos también un ejemplo de algo que no es geométrico. Vamos\(f\) be a polynomial in some number of variables. For now, we'll stick with 3 variables, \(x, y\), and \(z\). We say that \(f\) is a symmetric polynomial if every way of switching around (ie, permuting) the variables leaves \(f\) the same.

Por ejemplo, el polinomio\(f(x,y,z)=x+y+z\) is symmetric: switching the \(x\) and the \(z\), for example, gives \(z+y+x\), which is the same as \(f\). As a more complicated example, you can check that \(g(x,y,z)=x^2y+x^2z + y^2x + y^2z +z^2x + z^2y\) is also symmetric.

Por otra parte,\(h(x,y,z)=x^3+y^3+z\) is not symmetric, since switching \(x\) and \(z\) produces \(z^3+y^3+x\), which is not equal to \(h\). This polynomial does have some symmetry, since switching \(x\) and \(y\) leaves \(h\) the same, but we save the name 'symmetric polynomial' for the fully symmetric polynomials.

Ejercicio 1.2.0:

Vamos\(f\) be a symmetric polynomial with \(n\) variables. how many symmetries does \(f\) have?

Si no has probado un problema como este antes, trabajando en\(n\) variables - it is extremely important to get some practice. Try writing down some different symmetric polynomials with small numbers of variables. Is there a formula that describes the the number of symmetries in terms of the number of variables?

Los polinomios simétricos son cosas realmente interesantes, ¡y los volveremos a ver cuando hablemos de anillos y espacios vectoriales!

Colaboradores y Atribuciones

Template:ContribDenton