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# 1.3: Polinomios simétricos

• • Tom Denton
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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!