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Results tagged with quadratic-forms
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user 10898
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.
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On the image of a polynomial map modulo two distinct primes
Let $Q_0, Q_1, Q_2 \in \mathbb{Z}[x_0, x_1, x_2]$ be three non-singular ternary quadratic forms with integer coefficients. Let $T$ be a large real number, and let $p, q$ be two distinct primes having …
- 25k
3
votes
1
answer
104
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The density of diagonal isotropic ternary quadratic forms with respect to discriminant
Let $q(x,y,z) = ax^2 + by^2 + cz^2$ be a non-singular diagonal ternary quadratic form with integer coefficients. The discriminant $\Delta(q)$ of $q$ is then equal to $abc$, and for any positive number …
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A question on vectors in $\mathbb{R}^4$
Let $M$ be an invertible $4 \times 4$ real matrix, and let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^4$. Consider the matrix
$$\displaystyle \mathcal{M} = \mathcal{M}(\mathbf{u}, \mathbf{v}) = [\mathbf{u …
- 25k
1
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1
answer
315
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Counting integral points on a diagonal conic
Let $q(x,y,z) = x^2 - by^2 - cz^2$ where $b,c$ are co-prime positive integers. Suppose that the binary quadratic form $f(x,y) = x^2 - by^2$ is irreducible. I am interested in counting integral points …
- 25k
5
votes
1
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125
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Evaluating a binary quadratic form at convergents
We use the notation
$$\displaystyle [a_0; a_1, \cdots, a_n] = a_0 + \cfrac{1}{a_1 + \cfrac{1}{\ddots + \cfrac{1}{a_n}}}$$
to denote a finite continued fraction, and for a given real number $\alpha$, …
- 25k
2
votes
1
answer
329
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Representation of two related integers by the same binary quadratic form
Let $f(x,y) = ax^2 + bxy - cy^2$ be an indefinite, irreducible, and primitive binary quadratic form. That is, we have $\gcd(a,b,c) = 1$ and $\Delta(f) = b^2 - 4ac > 0$ and not equal to a square intege …
- 25k
1
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1
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Solving a pair of ternary quadratic form equations
Let $Q_1(x_0, x_1, x_2), Q_2(x_0, x_1, x_2) \in \mathbb{Z}[x_0, x_1, x_2]$ be two primitive, non-singular ternary quadratic forms (possibly indefinite). Suppose we want to solve the simultaneous equat …
- 25k
1
vote
0
answers
147
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When does one quadratic form divide another?
Let $Q_1, Q_2$ be two quadratic forms with integer coefficients in 4 variables $x_1, x_2, x_3, x_4$, both non-singular and not proportional. For a positive number $X$, which we may assume to be large …
- 25k
4
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1
answer
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Syzygy between covariants of pairs of ternary quadratic forms
In the book Nonlinear Computational Geometry, Page 208 (or page 15 of the online version on the author's website: http://www.loria.fr/~petitjea/papers/imaconics.pdf), Remark 5.1, Petitjean states that …
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5
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207
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Linearly independent quadratic forms vanishing on a finite set of points
The question I am interested in can be summed up as follows: given positive integers $n,m,k$, how do we write down $m$ linearly independent quadratic forms $Q_1, \cdots, Q_m \in \mathbb{C}[x_0, \cdots …
- 25k
5
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1
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182
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Stabilizers of pairs of ternary quadratic forms
Let $A,B$ be two ternary quadratic forms with real coefficients, given by symmetric matrices
$$\displaystyle 2A = \begin{pmatrix} 2a_{11} & a_{12} & a_{13} \\ a_{12} & 2a_{22} & a_{23} \\ a_{13} & a_ …
- 25k
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Counting 'admissible' binary quadratic forms
Let $f(x,y) = f_2 x^2 + f_1 xy + f_0 y^2$ be a primitive, positive definite, and reduced binary quadratic form. Put $k_f$ for the fundamental discriminant associated to $f$. That is, $k_f$ is square-f …
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4
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3
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On the automorphism group of binary quadratic forms
This question is a continuation of the following two questions:
Discriminants of indefinite integral binary quadratic forms admitting 3 or 6 torsion.
On certain solutions of a quadratic form equatio …
- 25k
3
votes
1
answer
263
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Equivalence of binary quadratic forms over $\operatorname{GL}_2(\mathbb{Q}_p)$ or $\operator...
Let $f(x,y) = ax^2 + bxy + cy^2$ be a binary quadratic form with integer coefficients and non-zero discriminant $\Delta(f)$. It is well-known that $f$ is $\operatorname{GL}_2(\mathbb{R})$-equivalent ( …
- 25k
4
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1
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181
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Conics and triples of binary quadratic forms
Let $C \subset \mathbb{P}^2$ be a planar conic curve, defined by a ternary quadratic form $Q(x_1, x_2, x_3)$ say. Suppose that $C(\mathbb{Q}) \ne \emptyset$, or equivalently, that $C$ is everywhere lo …
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