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Suppose I want to know what is known about the following polynomial $$x^4 - 2x^2y^2 - 2x^2z^2 + y^4 - 2y^2z^2 + z^4.$$

Where can one find such information?

Michael Rozenberg
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DSblizzard
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    What kind of information do you want? – ajotatxe Aug 13 '17 at 09:19
  • i have found this here https://www.google.at/url?sa=t&rct=j&q=&esrc=s&source=web&cd=3&cad=rja&uact=8&ved=0ahUKEwi2rca18NPVAhWKtxQKHc2hDHUQFgg4MAI&url=http%3A%2F%2Fhomepages.math.uic.edu%2F~jan%2Fdemo.html&usg=AFQjCNGOCkN6hDVox34zWaIJBdPKlyHYdQ – Dr. Sonnhard Graubner Aug 13 '17 at 09:21
  • @ajotatxe: for example, maximal number of facets in chambers of complement of corresponding hyperplane arrangement. But this example suits only restricted set of polynomials, so corresponding information should be different for each polynomial, of course. – DSblizzard Aug 13 '17 at 09:25
  • @Dr. Sonnhard Graubner: It is too specialized – DSblizzard Aug 13 '17 at 09:28
  • Related question: https://math.stackexchange.com/questions/1148371/is-there-some-database-or-software-to-look-for-patterns-in-polynomials. It for some reason wasn't shown while I typed mine. – DSblizzard Aug 13 '17 at 09:30

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It's just $$(x+y+z)(x-y-z)(x+y-z)(x-y+z).$$ We can get it by the following way. $$x^4+y^4+z^4-2x^2y^2-2x^2z^2-2y^2z^2=$$ $$=x^4+y^4+z^4-2x^2y^2-2x^2z^2+2y^2z^2-4y^2z^2=$$ $$=(x^2-y^2-z^2)^2-(2yz)^2=$$ $$=(x^2-y^2-z^2-2yz)(x^2-y^2-z^2+2yz)=$$ $$=(x^2-(y+z)^2)(x^2-(y-z)^2),$$ which gives the final factorization.

Also, the area of a triangle with sides-lengths $x$, $y$ and $z$ it's $$\frac{1}{4}\sqrt{2x^2y^2+2x^2z^2+2y^2z^2-x^4-y^4-z^4}$$

Michael Rozenberg
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