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I have the polynomial $6h^2+8k^2+12l^2-12hk-12hl+16kl$ and I would like to factorise it to find the signature of the quadratic form. The correct answer is $6(h−k−l)^2 +2(k+l)^2 +4l^2$, which means that the signature is positive-definite.

My attempt got me to $6(h-k)^2+6(h-l)^2+8(k+l)^2-6h^2-6k^2-2l^2$, but I can't get further towards the correct answer.

Is there a systematic way to do this?

Filip Nilsson
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1 Answers1

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Yes, there is a systematic way: first of all write it as $$a(h+rk+sl)^2+\phi(k,l)$$ where $\phi$ is a quadratic form in $k$ and $l$. Then write $$\phi(k,l)=b(k+rl)^2+\psi(l).$$

Angina Seng
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  • Hmm, thanks! Is there a name or something for this? Is it obvious that $\phi$ should be in $k$ and $l$? Does a similar approach work if the original form is in two variables? – Filip Nilsson May 27 '17 at 14:25
  • The point is to choose $r$ and $s$ to ensure that $h$ doesn't appear in $\phi$. – Angina Seng May 27 '17 at 14:29