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Results tagged with quadratic-forms
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user 29241
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.
2
votes
Accepted
''Local-global-principle'' for certain isometries of lattices
Would the last two pages of
http://wkchan.web.wesleyan.edu/qflecturenotes.pdf
help?
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5
votes
Accepted
Quadratic forms and $p$-adic integers
Let $V$ be the $n$-dimensional quadratic space over $\mathbb Q_p$ corresponding to the sum of $n$ squares. The matrix $M$ can be viewed as the Gram matrix for a $\mathbb Z_p$-lattice $L$ on $V$. You …
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3
votes
What's in the genus of the cubic lattice?
Both statements are well-known in the arithmetic theory of quadratic forms (aka integral lattices).
The first statement for $n \leq 5$ is a consequence of Hermite's bound on the nonzero minimum of …
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5
votes
Accepted
genus 2 Siegel theta series of 3-dimensional lattices
It is known, by Kitaoka's theory of characteristic sublattices, that if two lattices of rank $n$ with the same discriminant representing the same collection of lattices of rank $n - 1$, then the two l …
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4
votes
How to determine $O(L)$ is finite or not?
Maybe it is too late. A lattice has a finite orthogonal group if and only if it is definite or it is on the hyperbolic plane. This can be deduced from Satz (30.4) in Kneser's book Quadratische Forme …
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11
votes
Accepted
Integer positive definite quadratic form as a sum of squares
This is a well-known problem, called the Waring's problem of integral quadratic forms. Every semi-positive definite quadratic form in $n \leq 5$ variables is a sum of $n + 3$ squares of linear forms. …
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1
vote
Integer positive definite quadratic form as a sum of squares
I should add that there have been many results on this problem since the 1990s. The existence of the number $g(n)$ was established by Maria Icaza in her Ohio State 1992 thesis. You should check out …
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1
vote
Methods to decide whether two positive definite ternary quadratic forms are in the same spin...
Magma can determine the spinor genera in a given genus (of course, within the range of forms that it can handle).
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1
vote
On the orthogonal group of a lattice on a quadratic space over dyadic local field
I don't think that it is true for dyadic local fields. But I think in that case $O(L)$ is generated by symmetries and Eichler-transformations. Try the paper ``Generation of integral orthogonal group …
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6
votes
Accepted
orbits of automorphism group for indefinite lattices
It is in Kneser's book Quadratische Formen. For each r, there are only finitely many classes of representations of r by the lattice.
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1
vote
Bounded version of linear and quadratic Hasse--Minkowski theorem
For the first question, the answer is yes; you can find the statement and a proof in Cassels (pp 86-89). A far reaching generalization was obtained by J. Vaaler in 1987 concerning the height of a tot …
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5
votes
"Pythagoras number" for integral matrices
In Maria Icaza's Ohio State PhD thesis (1992?) she established the following result. For any positive integer $n$, let $S(n)$ be the set of $\mathbb Z$-lattices (or integral quadratic forms, if one p …
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