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Question

I am interested in the root of the polynomial function :

$x^4+(a+b+c+d-2)x^3+(ab+ac+ad+bc+bd+cd-2b-2c-a-d)x^2 +(abc+abd+acd+bcd-ab-ac-ad-2bc-bd-cd-a-d+b+c)x+ abcd-abc-bcd-ad+bc=0$

Under the restriction that: $0<a,b,c,d<1$.

I tried Vietas formula but don’t know how to come to a decomposition

$a_0=abcd-abc-bcd-ad+bc=x_1x_2x_3x_4$

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    $\begingroup$ Have you tried separating into multiple simpler polynomials? You can easily see that parts of the terms come from the decomposition of $(x+a)(x+b)(x+c)(x+d)$... From where does this question come from? $\endgroup$
    – Beni Bogosel
    Commented Apr 19, 2022 at 14:15
  • $\begingroup$ I tried it and first thought that $(-b)(-c)(1-a)(1-d)$ is the decomposition, but it’s not correct, there are some additional terms in my function. Now i have no idea for another decomposition which fits. $\endgroup$
    – toni_iva
    Commented Apr 19, 2022 at 14:43
  • $\begingroup$ This is a characteristic function of a matrix and i search for which $x$ the determinant is 0. $\endgroup$
    – toni_iva
    Commented Apr 19, 2022 at 14:45
  • $\begingroup$ Quartic equations are solved in radicals - e.g., see mathworld.wolfram.com/QuarticEquation.html $\endgroup$
    – Max Alekseyev
    Commented Apr 19, 2022 at 14:52

1 Answer 1

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There are two real solutions, the general expressions are very, very long. Even if all four coefficients are equal, $a=b=c=d$, you have an equation $$a^4-2 a^3+(4 a-7) a^2 x+(4 a-2) x^3+6 (a-1) a x^2+x^4=0$$ with complicated solutions.

For example, if all coefficients are equal to 1/2 the two real solutions are $$x=\frac{4+6^{2/3}\pm\sqrt{20 \sqrt{4+6^{2/3}}-6 \sqrt[3]{6}+4\ 6^{2/3}+32}}{4 \sqrt{4+6^{2/3}}}.$$

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  • $\begingroup$ Thank you for the comment. How do you know there are two real solutions? $\endgroup$
    – toni_iva
    Commented Apr 19, 2022 at 15:31
  • $\begingroup$ I need the general expression. How can i obtain it? $\endgroup$
    – toni_iva
    Commented Apr 19, 2022 at 15:39
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    $\begingroup$ Mathematica will readily give you the general expression for all four roots (two real, two complex); the expression for each root is about 250 lines long, what could you possibly do with it? $\endgroup$
    – Carlo Beenakker
    Commented Apr 19, 2022 at 16:04
  • $\begingroup$ I just tried with Mathematica, but I always get an error. :( $\endgroup$
    – toni_iva
    Commented Apr 19, 2022 at 16:36
  • $\begingroup$ Do you solve it with ‚Roots‘ or with ‚Reduce‘? $\endgroup$
    – toni_iva
    Commented Apr 19, 2022 at 16:47

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