Questions tagged [quadratic-forms]
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.
543 questions
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How should the first n natural numbers be arranged in a circle to minimize the sum of the products of adjacent pairs? [closed]
I was able to find (and prove) arrangements that would result in the sum of the products of adjacent pairs attain the maximum.
I am able to conjecture that the arrangement that would result in the ...
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Siegel's formula for generalized theta series with characteristics?
Siegel's formula(Siegel-Weil) directly relates the weighted sum of theta functions to Eisenstein series. (Or equivalently, the weighted sum of the cusp form is zero). I wonder if there is a ...
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Stabilizer group uniquely determines subspace
Let $(Q,V)$ be a quadratic space over an algebraically closed field $k$.
Let
$$ SO_Q(k):= \{ \sigma \in GL(V) : Q(\sigma v) = Q(v) \ \text{for all} \ v \in V \ \text{and} \det(\sigma) = 1 \}$$
Let $L \...
- 63
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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 ...
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How to solve a system of quadratic equations?
Suppose we have a system of $p$ quadratic equations about $\mathbf{x} \in \mathbb{R}^3$ and $\mathbf{x} > 0$
$$ \left\{
\begin{array}{lr}
\mathbf{x}^\top \mathbf{C}_1 \mathbf{x} = 1, ...
- 19
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Modified straightline complexity of almost square of sums
Assume every linear operation (such as inner product with constant vector) can be performed in one step and multiplication by variables (quadratic operation) can be performed in one step.
We know the ...
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Necessary and sufficient condition for a prime to be represented by an arbitrary positive definite binary quadratic form $ax^2+bxy+cy^2$
Given an arbitrary (but fixed) positive definite primitive integral binary quadratic form $g(x, y)=ax^2+bxy+cy^2$, and let $m$ be an arbitrary integer. We will denote the discriminant of $g$ by $D=D_g=...
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Maximizing quadratic forms
Consider the maximization problem
$$\text{maximize} \quad Q(x)= \sum_{i<j} \Big(\sum_{k} a_{ik}a_{jk}\Big) x_i x_j \quad \text{subject to} \quad \sum_{i}x_i^2=1,$$
and let $M$ be maximum value ...
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Computing local residues of traces of bilinear forms on algebraic number fields
This is a question about computing the local residues of a general symmetric bilinear form over $\mathbb{Q}$. I've been using Lam's Introduction to Quadratic Forms as a reference, but I'm stuck on the ...
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When does one quadratic form divide another?
Let $Q_1, Q_2$ be two quadratic forms with integer coefficients in 4 variables $x_1, x_2, x_3, x_4$, both non-singular and not proportional. For a positive number $X$, which we may assume to be large ...
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Proper ideals are invertible
I am reading through Cox's book Primes of the form $x^2+ny^2$ and I am stuck with some proofs in Chapter 7 (I have the 2nd edition). There, the author presents the following Lemma:
Lemma 7.5: Let $...
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Norm quadrics and their motives
Let $k$ be a field of characteristic $\neq 2$ and $\langle\!\langle a_{1},\cdots,a_{n}\rangle\!\rangle$ a Pfister form over $k$. Denote by $Q_{\underline{a}}=Q_{a_{1},\cdots,a_{n}}$ the projective ...
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On universal sums $x(ax+b)/2+y(cy+d)/2+z(ez+f)/2$ over $\mathbb N$
Let $a,b,c,d,e,f$ be integers with $a\ge c\ge e>0$, $b>-a$ and $a\equiv b\pmod2$, $d>-c$ and $c\equiv d\pmod 2$, $f>-e$ and $e\equiv f\pmod2$. If each $n\in\mathbb N=\{0,1,2,\ldots\}$ can ...
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How explicitly write a projective transformation between the conics over the univariate function field?
Consider the quadratic forms
$$
Q_1 = x^2 + y^2 - (t^2+1)z^2,\qquad Q_2 = x^2 + y^2 - z^2
$$
over the rational function field $\mathbb{F}_p(t)$, where $p > 2$ is a prime such that $t^2 + 1$ is ...
- 1,833
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Reference request: Number of elliptic and hyperbolic quadratic forms of a given rank over a finite field
My question is over the finite field $\mathbf{F}_q$ of $q$ elements. It is well known that a symmetric matrix of odd rank corresponds to a parabolic quadratic form but even rank symmetric matrices ...
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Question about Notation for Spaces of $n$-ary $k$-ic Forms
Define an $n$-ary $k$-ic form to be a polynomial over the integers of homogeneous degree $k$ in $n$ variables. In Section 1 of the paper "Higher Composition Laws I" (linked below), Bhargava writes $(\...
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Isotropy of skew-Hermitian forms over division algebras
Assume char(F) $\neq$ 2.
Let $D$ be a central division algebra over a field $F$ and $h: V \rightarrow D$ be an anisotropic skew-Hermitian form. We can easily see that $h_{\bar{F}}$ is totally ...
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Showing a modified system of quadratic equations is stable
I have and $n$ dimensional dynamical system, given by
$\dot{x} = M D(x) P x - \frac{c}{2}x$
$P$ is a full rank $n \times n$ matrix, with $p_{ij} \in [0,c]$, such that $p_{ij}=c-p_{ji}$ for some ...
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Are there any good references to modular form and its application to quadratic forms
I am a beginner of this interesting branch of mathematics，I have read N.Koblitz's Introduction to Ellipitic Curves and Modular Forms.I am familiar with the arithmetic theory of quadratic forms.Thanks ...
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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-...
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Generalized eigenvalue problem with nonnegative eigenvector constraint
Consider the following problem that is known to be non-convex but can be solved as a generalized eigenvalue problem (i.e. has a global optimum solution):
$\underset{w}{\text{maximize}}\quad w^{\top}...
- 11
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Conditions on $\beta$ under which the trace pairing restricted to $\mathfrak{so}(V,\beta)$ is positive (negative) definite
Let $V$ be a finite dimensional vector space over $ \mathbb{R}$. Let
\begin{equation} \left\langle\:,\:\right\rangle:\mbox{End}(V)\otimes\mbox{End}(V)\rightarrow \mathbb{R}\end{equation}
denote the ...
user70004
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Is the genus symbol implemented?
Conway and Sloane, as well as Cassels, and also O'Meara, all have their own idiosyncratic way of expressing the following result (for good primes): Every quadratic form with coefficients in $\mathbb{...
- 2,429
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Is this QCQP convex or nonconvex?
\begin{equation}
\begin{split}
\min_{x\in \mathbb{R}^n}\:f(x)=(1/2)x^{T}Q_0x+c_0^T x
\end{split}
\end{equation}
s.t.
$$
g_i(x)=\frac{1}{2}x^T Q_ix-lmax_i\leq0,i\in\{1,...,m/2\}
$$
$$
g_i(x)=\frac{...
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Statements generalizing representability of integers by binary quadratic forms to $n$-variable higher homogeneous forms?
Representing integers through the theory of binary quadratic forms is a well studied topic. We know that given $a,b,c\in\Bbb N$, based on discrimant $b^2-4ac$, we can study the representability of ...
user76479
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Bound of spectral radius of polynomial of a complex matrix
I am trying to prove or disprove the following inequality.
$$ ||P(A)||_2\leq 2 \max_{\alpha\in W(A)}| P(\alpha)|,$$
where $P(\cdot)$ is a complex polynomial, $A\in \mathbb{C}^{n\times n}$ and $W(A)...
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Minimal Length of Quadratic Forms
Let $Q(x_1,\dots,x_n)=X'PX$ be a quadratic form with all non-negative and integral coefficients given by $$Q(x_1,\dots,x_n)=\sum_{i=1}^cf_i^+(x_1,\dots,x_n)g_i^+(x_1,\dots,x_n)$$ where $c$ is the ...
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Cassels-Birch-Davenport theorem for multiple quadratic forms of certain type
A classical theorem of Cassels states that if a homogenous quadratic form $Q$ has an integer zero, then there is a zero of small height (bounded solely by the coefficients and number of variables). ...
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Is this related to a simple property of a lattice?
I am looking for a certain notion of sparseness of lattices.
I want to find a vector in $\mathbb{Z}^N$ that the minimal possible inner product with all the vectors of a given lattice. Or at least, I ...
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Tensor product with $\mathbb{R}$ of an even unimodular lattice
Let $\Lambda$ be an unimodular even lattice of signature $(m,n)$.
By a classifying theorem by Milnor, $\Lambda$ must be of the form $U^k\oplus E_8(\pm 1)^l$, where $U$ is the hyperbolic plane.
Now ...
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Orthogonal transformations with trivial spinor norm as product of reflections $r_w$ with $(w,w)=-2$
I'm trying to prove that, for a standard unimodular even lattice $\Lambda$ (by standard I mean that it is direct sum of copies of the hyperbolic plane $U$ and $E_8$) every element of $O^+(\Lambda)$, i....
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Non-negative Quadratic forms with Exterior Forms
Hello All,
I apologize if the following question is too elementary. Any suggestion is greatly appreciated. Thank you.
Let $n\geqslant 4$, $X$ be an $n$-dimensional inner product space over $\mathbb{...
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Traversing perfect quadratic forms
I am wondering whether there is an efficient algorithm to traverse all the $N\times N$ perfect quadratic forms $Q$ inside the polyhedron $e_j^T Q e_j \geq 1$, $j = 1\ldots m$, where $e_j$ are some ...
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Watson Transformation Squared reference request
See
http://www.numbertheory.org/obituaries/OTHERS/watson.html
George Leo Watson (1909-1988) wrote in a conversational manner, it is difficult to see when he switches from the trivial to the ...
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rank of outer product
I would like to ask if you may know how to prove this claim or any theorem related:
Given 9 points (x,y,z) lie on unit sphere in 3 dimensional space such that any 4 points are not on the same plane. ...
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Difference of two sums of two squares [closed]
Is there a way to prove that if n = (a^2 + b^2) - (c^2 + d^2), then for every natural n, there are infinitely many a, b, c and d?
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About list of discriminants of real quadratic fields with narrow class number 1?
I have a couple of questions regarding the list of discriminants of real quadratic fields with narrow class number 1.
The sequence A003655 in OEIS portraits a list of discriminants of real quadratic ...
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two non-degenerate quadratic forms on $GF(2)^2r$
I know this:
There are two non-degenerate quadratic forms on $GF(2)^2r$. The hyperbolic form may be taken to be
$Q^+(x)=x_0 x_1 + \cdots +x_{2r-2}x_{2r-1}$ ,
and the elliptic form to be
$Q^-(x)=x^...
- 179
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Integers representable as binary quadratic forms
It is known that odd prime $p$ can be represented as $p=x^2+y^2$ if and only if $p \equiv 1$ mod $4$, represented as $p=x^2+2y^2$ if and only if $p \equiv 1$ or $3$ mod $8$, represented as $p=x^2+3y^2$...
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Maximum dimension of a simultaneous anisotropic subspace of quadratic forms over $ \mathbb{Q} $
Let $(V,q )$ be a quadratic space over $ \mathbb{Q} $. A subspace $ U $ is called totally isotropic if $ q(x) = 0 $ for all $ x \in U $ and a subspace $ U $ is called an anisotropic subspace if $ ...
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Vanishing bilinear forms
For a symmetric or antisymmetric bilinear form $\varphi$ on a vector space $V$, if $\varphi(x,y)=0$ then also $\varphi(y,x)=0$ ($x,y\in V$).
I was wondering if this is also a necessary condition for ...
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Nondegenerate projective plane conics over a nonperfect field $k$ of even characteristics
Let $C_1\!: a_1x^2 + b_1y^2 + c_1z^2 + d_1xy + e_1xz + f_1yz$ and $C_2\!: a_2x^2 + b_2y^2 + c_2z^2 + d_2xy + e_2xz + f_2yz$ be nondegenerate projective plane conics over a nonperfect field $k$ of even ...
- 1,833
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Name for a Specific Type of Non-Symmetric Bilinear Form
Let $V$ be a finite dimensional vector space, with some choice of basis $\{e_i\}_{i \in I}$. With respect to an idempotent bijection $B:I \to I$, define a bilinear form by
$$
g = \sum_{i=1}^N \lambda_{...
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2
answers
194
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Quadratic subextension of the function field of quadric.
Assume $F$ is a field of characteristic $\neq 2$. Let $(V,q)$ be a quadratic space such that $\rm dim~ q\geq 3$. When $q$ is irreducible it is known that
there exist a purely transcendental field ...
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235
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Is there a name for a formula to calculate ascending numbers to a quadratic-like sequence?
For e.g. any range of number 0 - n
0 1 2 3 4 5 6
to:
0 2 4 6 4 2 0
Is there a name for this kind of formula or calculation?
- 3
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127
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Automorphism groups in class sets of ternary lattices
Let $\Lambda$ be an integral lattice in some definite ternary quadratic space $(V,Q)$ over $\mathbb{Q}$.
Consider the usual class set $\text{Cl}(\Lambda) = O(V)\backslash\text{Gen}(\Lambda)$, i.e. the ...
- 562
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Compatibility conditions for quadratic equations
In the context of physics, I stumbled over the following problem: I have $N$ equations, all are quadratic in a single scalar, real variable $x$:
\begin{eqnarray}
0 &= A_1x^2 + B_1x + C_1 \\
&...
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1
answer
125
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Polyhedral conditions for quadratic inequalities in fixed dimension
Denote $\mathcal T$ be set of $(T_1,T_2,T_3,T_4)\in\mathbb Z^4$ that satisfy
$$0<T_1,T_2,T_3,T_4$$
conditions?
Define the level set $$M_{\gamma}(Q,\mathcal T)=\{(T_1,T_2,T_3,T_4)\in\mathcal T:Q(...
- 13.9k
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Elementary description to count of perfect squares - II
What can we say about growth of smallest gap $g(a)$ which is the smallest $|x-y|$ where $0\leq x,y\leq\Big\lfloor\frac a2\Big\rfloor$ and $\sqrt{x(a-x)},\sqrt{y(a-y)}\in\Bbb Z$?
Is $g(a)=1\iff a=b^2+...
- 13.9k
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1
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Elementary description to count of perfect squares - I
Is there an elementary description of $$N(a)=\Big|\Big\{x\in\{0,1,\dots,\Big\lfloor\frac a2\Big\rfloor-1,\Big\lfloor\frac a2\Big\rfloor\Big\}:\sqrt{x(a-x)}\in\Bbb Z\}\Big|$$ and though likely non-...
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