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Results tagged with quadratic-forms
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user 14830
Algebraic and geometric theory of quadratic forms and symmetric bilinear forms, e.g., values attained by quadratic forms, isotropic subspaces, the Witt ring, invariants of quadratic forms, the discriminant and Clifford algebra of a quadratic form, Pfister forms, automorphisms of quadratic forms.
3
votes
Reflections on affine quadric hypersurfaces
Not necessarily, not even if $f$ is negative definite:
then $X$ is a Euclidean root system with all norms equal,
and $R$ is its Weyl group, which is transitive iff
the root system is irreducible.
The …
- 79.9k
5
votes
Accepted
The density of diagonal isotropic ternary quadratic forms with respect to discriminant
The density must be zero: for each prime $p$,
the probability that $q$ is not $p$-adically isotropic is
$3/2p + O(1/p^2)$ (when one of $a,b,c$ is a multiple of $p$,
and the product of the other two …
- 79.9k
1
vote
Applications of isotropic quadratic forms
One standard construction of $L^2$ is to start from a space of
square-integrable functions, which is only positive semidefinite
(and thus contains many isotropic vectors), and form the quotient by
the …
- 79.9k
4
votes
Accepted
Stabilizers of pairs of ternary quadratic forms
It's actually order 8 for no real zeroes and order 4 for two real zeroes,
not the other way around. (See the bottom of page 1038 of
the paper.)
The symmetry groups are taken modulo $\{ \pm 1 \}$, s …
- 79.9k
8
votes
Accepted
What's in the genus of the cubic lattice?
[edited mostly to add information about $n > 9$]
David Treumann's guess is correct: ${\bf Z}^n$ is unique in its genus
iff $n \leq 8$, and for $n = 9$ the genus consists of only ${\bf Z}^9$ and
${\bf …
- 79.9k
3
votes
Accepted
2-dimensional sublattices with all vectors having very big square (in absolute value)
For the record: OP Misha Verbitsky writes that
"[$\Lambda$ of] rank $\geq 6$ or $\geq 7$ is a usual assumption
in these kind of applications", in which case
Ekaterina Amerik's suggestion of using
$N\ …
- 79.9k
11
votes
Accepted
minimizing an integral over integer-coefficient polynomials $\displaystyle \inf_{f \in \math...
Thanks to BigM for the link to Ofir's
MO Question 121913,
which cites a 120-year-old paper of Hilbert for the result that
the integral can get arbitrarily small as long as $b-a < 4$:
D. Hilbert: E …
CommunityBot
- 1
16
votes
Accepted
Positive primes represented by indefinite binary quadratic form
Class field theory promises such a polynomial (more properly,
such a number field $H$, since a polynomial generating $H$ might have to err
on the first few primes, though in our case it turns out ther …
CommunityBot
- 1
5
votes
Root systems and sums of squares
Just saw this thanks to a "Related" link from
Question 154928.
Yes, it is known that the $E_6$ form cannot be written as a sum of
integral squares, and thus (by specialization) that the same is true o …
CommunityBot
- 1
3
votes
Accepted
Are these powers of a characteristic 3 power series annihilated by certain Hecke operators?
Yes, and $D \equiv \eta(2z)^{12}$ certainly helps: it means
$$
D^{(1+3^m)/4} \equiv \eta(2z)^{3+3^{m+1}} \equiv \eta(6z) \, \eta(2\cdot 3^{m+1}z).
$$
Now $\eta = \sum_a \chi(a) q^{a^2/24}$ where $a$ r …
- 79.9k
16
votes
Accepted
How many solutions are there to $\sum_{i=1}^3 x_i^2+x_iy_i+y_i^2=k$?
The formula that emiliocba seeks seems to be as follows.
Let $\chi$ be the Dirichlet character mod $3$. For $k>0$
write $k = 3^e n$ with $n \equiv \pm 1 \bmod 3$. Then
the number of representations …
- 79.9k
8
votes
Accepted
Is the square of the covering radius of an integral lattice/quadratic form always rational?
Yes, the square $R^2$ of the covering radius is always rational; and in small dimensions its denominator is always a factor of $2^{n+1} \Delta$ where $\Delta$ is the lattice discriminant, but possibly …
- 79.9k