# 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.

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### Squares in skew fields of dimension 2 over a sub skew field

Let $\ell$ be a skew field (i.e., a division ring), and let $k$ be a sub skew field, such that the dimension of $\ell$ as a left vector space over $k$ is $2$. Then if $a \in \ell \setminus k$, we can ...
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### Sets represented by integral quadratic forms

Let $f(x) = x^\intercal A x$ be a positive definite integral quadratic form on $d$ variables. A positive integer $n$ is said to be represented by $f$ if $f(x) = n$ for some $x \in \mathbb Z^d$. A set ...
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### How to express a quadratic polynomial exactly as a power series [closed]

I claim, for $\operatorname{artanh}(\rho) = \frac{1}{2} \ln\left(\frac{1+\rho}{1-\rho}\right)$, i.e., the inverse hyperbolic tangent function, the following holds approximately under assumptions given ...
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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 ...
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### Low rank matrices which preserve maximizers of quadratic form

Suppose $x, y \in \{0,1\}^d$ are binary vectors. For a matrix $M$ consider the quadratic form, \begin{align} x^T M y + (\mathbb{1} - x)^T M (\mathbb{1} - y) \end{align} Does there exist an $M$ such ...
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### A possible variant of Zagier's one-sentence proof for Fermat's sum of two squares theorem?

Is it possible to modify Zagier's one-sentence proof of Fermat's sum of two squares theorem (see here) to prove certain non-trivial cases of Jacobi's four-square theorem (see here)? Let $p$ be a prime ...
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### Convergence in probability of quadratic form with positive mean

Let $\boldsymbol{X}_n\in\mathbb{R}^n$ be a sequence of Gaussian random vectors with independent entries, such that $X_{n,i}\sim \mathcal{N}(\mu_i,\sigma^2)$ (that is, all entries of the $n$th vector ...
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### Totally isotropic space for bilinear pairing over ring

A duplicate of this: Consider the following well-known inequality: Let $b$ be a non-degenerate symmetric bilinear pairing over a (finite-dimensional) $\mathbb F$-vector space $V$ and $W$ a totally ...
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### Is there a way to linearize matrix quadratic forms?

Say $x$ is a random vector in $\mathbb{R}^n$. Then, given a (deterministic) symmetric real positive definite matrix $A$, if we want to calculate the expectation of the quadratic form, we can use the ...
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### Positive quadratic polynomial

Let $S$ be solutions of a system of quadratic polynomials on $\mathbb{R}^n$. Suppose $q$ is another quadratic polynomial such that $q|_S\geqslant 0$. Is it possible to find a polynomial $\tilde q$ ...
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### Sufficient condition for pair of real quadrics to have real intersection

In the following, when I talk about the zero of a homogeneous polynomial I always mean a projective zero. Let $q$ be a real quadric. Then $q$ has a real zero if and only if $q$ has indefinite ...
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### An arithmetic problem involving a system of equations

Fix a positive integer $r$. Describe the solutions to the system of equations given by: $$$$\sum_{1\leq i\leq r}X_i^2\equiv0\pmod{X_k}(1\leq k\leq r)$$$$ Example: In the case ...
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### Is there any point in considering Form Rings when 2 admits an inverse?

In the study of quadratic spaces over general rings, there is a type of scalar which people consider called a Form ring $(R,\Lambda)$ relative to some anti-automorphism denoted $(-)^J:R\to R$ and ...
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Let $Q(x,y) = ax^2+b xy + cy^2$ be a primitive integral positive-definite quadratic form, with associated number field $K$. If $D=b^2-4ac$ is a fundamental discriminant, then it's well-known that $$\... • 381 4 votes 1 answer 193 views ### Schur multiplier of a Chevalley group of type D_5 \DeclareMathOperator\EO{EO}\DeclareMathOperator\SO{SO}\DeclareMathOperator\St{St}\DeclareMathOperator\Sp{Sp}This is sort of a follow up question to my post here regarding the commutator subgroup of ... • 403 4 votes 0 answers 182 views ### Is an orthogonal direct sum decomposition with respect to two quadratic forms necessarily unique up to isomorphism Consider two quadratic forms Q and P over a finite dimensional vector space V over a quadratically closed (or perhaps Pythagorean) field F. If V can be decomposed as V = V_1 \oplus V_2 \... • 4,873 2 votes 0 answers 34 views ### Criterion for unicity and existence of pre-image in multivariate cryptography Repost from math.stackexchange since no one could help me there and it concerns my research. I am reading Ding's Multivariate Public Key Cryptosystems and in the book the author explains the so-called ... • 173 6 votes 1 answer 510 views ### Computing a Commutator Subgroup \DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}I’m studying the group \O(5,5,\mathbb{Z}), the indefinite orthogonal matrices with integer entries. In particular, I ... • 403 0 votes 0 answers 52 views ### How to express the product of elements of a vector with quadratic form? x is a binary vector which means the elements in x are 0 or 1, and p is another vector with the same length. How to express the product of elements in p whose corresponding elements are 1 in ... 2 votes 0 answers 214 views ### Solutions to the quadratic matrix equation X A X^T = B Let A, B \in \mathbb{R}^{n \times n} be symmetric, positive-semidefinite, full-rank matrices. I would like to understand the set of X \in \mathbb{R}^{n \times n} which are themselves symmetric and ... 4 votes 1 answer 252 views ### Is there a good notion of kernels of quadratic forms on abelian groups? Let G be an abelian group and let q:G \to \mathbb{Q/Z} be a quadratic form, i.e. q(a)=q(-a) and b(x,y)=q(x+y)-q(x)-q(y) is a bihomomorphism. On vector spaces, when people speak about the ... • 1,703 1 vote 0 answers 136 views ### When does a system of homogeneous quadratic equations have integer solutions? I learned that in general, solving systems of quadratic Diophantine equations is a difficult problem. But I wonder if there are special (and non-trivial) types of systems that are easier to handle. ... • 11 7 votes 0 answers 243 views ### K3 surfaces with no −2 curves I seem to remember that a K3 surface with big Picard rank always has smooth rational curves. This question is equivalent to the following question about integral quadratic lattices. Let us call a ... • 9,115 0 votes 0 answers 177 views ### Points at which a polynomial becomes reducible Let n \geq 10 and set \mathbf{y} = (y_1,\ldots,y_n). Let Q_1(\mathbf{y}),\ldots,Q_5(\mathbf{y}) be non-zero quadratic forms with integer coefficients such that the cubic form x_1Q_1(\mathbf{y})+... 0 votes 1 answer 182 views ### 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 \\ &... 1 vote 1 answer 94 views ### Projectivity of the fundamental ideal of Witt groups Suppose k is a field. I wonder when the Witt ring of the quadratic forms \textbf{W}(k) has a projective fundamental ideal, which is the kernel of the rank modulo 2 morphism. Here I want a ... • 824 1 vote 1 answer 69 views ### A question about the sign of quadratic forms on nonnegative vectors Let M be a real square matrix of order n\ge 3. Assume that for every nonnegative vector \textbf{z}\in \mathbb R^n which has at lease one zero entry we have \textbf{z}^T M \textbf{z} \ge 0. Can ... 5 votes 1 answer 240 views ### Equivalence of quadratic forms over p-adic integers vs over localisation at p To discern whether two integral quadratic forms are equivalent over the p-adic integers, one can compute a Jordan decomposition at p and read off some invariants. Restricting to p\ne2 for ... • 323 1 vote 2 answers 348 views ### Integral solutions of quadratic equation 5 X² − 14 X⁢Y + 5 ⁢Y² = n Solve for all integers x and y the quadratic form 5 X² − 14 X⁢Y + 5 ⁢Y² = n for some integer n. I know that for some cases there are recurrence solutions, but I'm not sure how to solve these ... 2 votes 0 answers 56 views ### Equidistribution of lattice points on quadratic forms without certain values I have recently been studying some results about equidistribution of lattice points on positive definite quadratic forms Q with 3 or more variables. Concretely the article Duke and Schulze-Pillot, ... • 313 10 votes 1 answer 888 views ### How to describe all integer solutions to x^2+y^2=3z^2+1? The question is in the title. Here is a short motivation. The general quadratic Diophantine equation is$$ x^TAx+bx+c=0,  where $x$ is a vector of $n$ variables, $A$ is $n \times n$ matrix with ...
We are given a matrix $D \in \mathbb{Z}^{C \times C}$ of non-negative entries, an integer $k \geq 1$ and we need to maximize the quadratic form $x^T D x$ under some simple constraints. For all ...