2
$\begingroup$

There is a huge amount of research dealing with analysis of representation of integers by a quadratic polynomials only with terms of degree 2 (here is for example review of some known methods by J.Hanke: http://www.math.ubc.ca/~cass/siegel/hanke-ternary.pdf). There is a theory where we can expess corresponding theta series as a sum of Eisenstein series and cusp forms and then make some estimation on representation number as coefficients of theese series.

My question is the next: is it possible to extend these methods for analysis of representation of integers by quadratic quadratic polynomial with linear terms, especially when $n=3$ (for example, $m=5a^2+5b^2+5c^2+2a+2b+4c$)? My knowledge of theory of Modular Forms is not so good, so I would ask you an advice. Thank you in advance.

Improve this question
$\endgroup$
5
  • 1
    $\begingroup$ Completing the squares doesn't work? $\endgroup$
    – Gerry Myerson
    Commented Feb 27, 2017 at 22:57
  • 1
    $\begingroup$ The term "quadratic form" means a homogeneous quadratic polynomial, i.e. one without any linear and constant terms. Your question concerns quadratic polynomials, not quadratic forms. $\endgroup$
    – GH from MO
    Commented Feb 27, 2017 at 23:11
  • $\begingroup$ Sorry, that's my lag in definition. $\endgroup$
    – Pavel Kozlov
    Commented Feb 28, 2017 at 5:38
  • $\begingroup$ What are $n$, $m$, $a, b, c$? $\endgroup$
    – Kimball
    Commented Feb 28, 2017 at 20:33
  • $\begingroup$ $m$ is an integer that we want to represent, $a,b,c$ are unknown integer variables. $\endgroup$
    – Pavel Kozlov
    Commented Feb 28, 2017 at 20:38

1 Answer 1

Reset to default
5
$\begingroup$

Yes, it is possible to extend these methods, although the picture is somewhat less clear than in the quadratic form case. When one discusses representations by a quadratic polynomial, this is equivalent to asking for representations by a coset of a lattice.

The theta series of a lattice coset is still a modular form, although usually for a smaller group than $\Gamma_{0}(N)$. This is described well in Iwaniec's book "Topics in Classical Automorphic Forms".

There is also a notion of genus for a lattice coset (see this paper of Wai Kiu Chan and Byeong-Kweon Oh), and this presumably says something about the Eisenstein series part of that theta series, although I don't think there's an analogue that's been worked out of the beautiful formula of Siegel that represents the Eisenstein series coefficient as a product of local densities. (At least, this is the impression I got from Wai Kiu Chan when I talked to him about it.)

Note that Chan has had a few students work on some problems about ternary quadratic polynomials (although not much from the modular forms perspective yet); in particular take a look at the work of Anna Haensch (see this paper and this paper) and James Ricci.

In some cases, the problem really is as simple as completing the square. (Exercise: Prove that every non-negative integer can be written in the form $a^{2} + b^{2} + c^{2} + ab + ac + bc + a + b + c$.)

Improve this answer
$\endgroup$
2
  • $\begingroup$ Jeremy, big thanks for your useful links! As for my example, completing the square does not work, because we have equation $5m+6=(5a+1)^2+(5b+1)^2+(5c+2)^2$, and elementary means do not clarify situation in that case, in contrary to, for example, the representation as sum of three triangular numbers. $\endgroup$
    – Pavel Kozlov
    Commented Feb 28, 2017 at 5:53
  • 1
    $\begingroup$ Here's one more link. The paper by Shigeaki Tsuyumine here is about writing numbers as sums of three squares where congruences conditions are put on those squares. The main results of that paper imply that for sufficiently large $m$, there is a representation of $5m+6$ as $(5a+1)^2 + (5b+1)^2 + (5c+2)^2$. $\endgroup$
    – Jeremy Rouse
    Commented Feb 28, 2017 at 14:20

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .