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I have read (probably) in Kanigel's book The Man Who Knew Infinity that S. Ramanujan devised his own method of solving the Quartic Equation after he learnt to solve the Cubic Equation. Does anyone know what exactly was Ramanujan's method of solving the Quartic Equation?


Note

I have previously posted the question here.

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    $\begingroup$ Meta discussion. $\endgroup$ – HDE 226868 Dec 7 '14 at 13:18
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    $\begingroup$ Note to all: The question on Mathematics was closed two hours ago. $\endgroup$ – HDE 226868 Dec 9 '14 at 0:42
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You can find it in Ramanujan's Notebooks IV by B. Berndt, Chap. 22 Elementary Results, Entry 20, p.31.

Ramanujan starts with this problem. Let $a,b,c,d$ be arbitrary. Solve the system,

$$x^2+ay = b\tag{20a}$$

$$y^2+cx = d\tag{20b}$$

Eliminating $y$, we find it is equivalent to,

$$a^2(d-cx) = (b-x^2)^2\tag{20.1}$$

Assume without loss of generality that $a=2$. Expanding this out,

$$x^4-2bx^2+4cx+(b^2-4d)=0\tag{20.1a}$$

By a simple linear substitution, the general quartic equation can be expressed in the depressed form,

$$x^4+px^2+qx+r = 0\tag{20.1b}$$

Equating coefficients of ${20.1a}$ and ${20.1b}$, you have a system of 3 equations in 3 unknowns {$b,c,d$}. Hence every quartic can be expressed in the form ${20.1}$. The problem then is to find $x$. Ramanujan defines,

$$x = \alpha+\beta+\gamma\tag{20.1c}$$

$$y = -(\alpha\beta+\beta\gamma+\gamma\alpha)$$

$$-c/2 = \alpha\beta\gamma$$

(If you are familiar with cubic equations, you'll already see where Ramanujan is going.)

Substitute $x,y,c$ into $(20a)$ and $(20b)$ keeping in mind $a=2$, then,

$$x^2+ay = \alpha^2+\beta^2+\gamma^2= b$$

$$y^2+cx = (\alpha\beta)^2+(\beta\gamma)^2+(\gamma\alpha)^2 = d$$

$$(-c/2)^2 = (\alpha\beta\gamma)^2$$

By elementary symmetric polynomials, we then conclude that $\alpha^2,\,\beta^2,\,\gamma^2$ are roots of the cubic equation,

$$t^3-bt^2+dt-c^2/4=0\tag{20.1d}$$

Of course, by solving $(20.1d)$, one can then find $\alpha,\,\beta,\,\gamma$. Using $(20.1c)$, we then further conclude that the four roots of the quartic are,

$$x = \alpha+\beta+\gamma, \quad\alpha-\beta-\gamma, \quad-\alpha-\beta+\gamma, \quad-\alpha+\beta-\gamma$$

P.S. This is similar to Euler's method where he solves a quartic as $x_i = \sqrt{y_1}\pm \sqrt{y_2}\pm \sqrt{y_3}$ and the $y_i$ are roots of a cubic. There's actually a generalization to this for solvable 8th deg eqns as $x_i = \sqrt{y_1}\pm \sqrt{y_2}\pm\dots\pm \sqrt{y_7}$ and the $y_i$ are roots of a 7th deg eqn. See this mathoverflow post.

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    $\begingroup$ @Adhvaitha (also Tito Piezas III) FYI, the Math.SE question and answer can be found here. Neither of these answers appear to be plagiarizing the other, so I hope nobody else downvotes. Tito, why did you post this here? The OP asked it here to get different answers; yours isn't anything new. $\endgroup$ – HDE 226868 Dec 5 '14 at 16:43
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    $\begingroup$ [Chain of comments deleted]. I would like to remind you all that this is not the place for (angry) discussions, certainly not about things that have very little to do with the history of science and mathematics. If you nevertheless want to talk about something, please do so in History of Science and Mathematics Chat. Here you can also reach me, or one of the other moderators. $\endgroup$ – Danu Dec 8 '14 at 21:53

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