Hi I have the following problems concerning quadratic and ternary forms. Any help would be greatly appreciated.

  1. $3\displaystyle\sum_{x, y\in\mathbb{Z}}q^{x^2+xy+7y^2}=3\displaystyle\sum_{x, y\in\mathbb{Z}}q^{9(x^2+xy+y^2)}+P_{3,1}\left(\displaystyle\sum_{x,y\in\mathbb{Z}}q^{x^2+xy+y^2}\right)$, where $P_{3,1}$ is the operator which takes only the exponents of $q$ which are congruent to 1 modulo 3.

  2. Which of the integers are not represented by the ternary form (5, 8, 11, -4, 1, 2) assuming that it is regular, though its not yet known whether its regular or not. Its genus has 2 elements.

  3. If $F$ is a ternary positive quadratic form with Hessian, $H(F)=5$. Then min (F)=1.

Comment: Though the upper bound for such minimum value turns out to be 2 inclusive, I am not able to eliminate the case when it is 2.

Any help would be greatly appreciated.



closed as off-topic by Will Jagy, Lucia, Chris Godsil, Yemon Choi, Ryan Budney Apr 24 '14 at 17:46

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    $\begingroup$ That's three questions that probably require three separate solutions. Better to ask them separately. Also, the notation $(5,8,11,-4,1,2)$ is too compressed for most of us to be sure what you meant: better to display the matrix or the quadratic polynomial you mean. $\endgroup$ – Noam D. Elkies Apr 23 '14 at 0:19
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    $\begingroup$ @Noam, i imagine this is homework. The form is 1620: $5 x^2 + 8 y^2 + 11 z^2 -4 y z + z x + 2 x y$ from my paper with Kaplansky and Schiemann. I put several lists as ordinary text files at zakuski.utsa.edu/~jagy because at least one of them is too large to email. $\endgroup$ – Will Jagy Apr 23 '14 at 1:13
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    $\begingroup$ @NoamD.Elkies, figured out who assigned these, cc'd you in the email $\endgroup$ – Will Jagy Apr 23 '14 at 2:55
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    $\begingroup$ In fact I have been informed this is from a take-home exam; no hints until tomorrow I am told. $\endgroup$ – Todd Trimble Apr 23 '14 at 4:00
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    $\begingroup$ This question appears to be off-topic because it is a question from an exam $\endgroup$ – Yemon Choi Apr 24 '14 at 13:43