Questions tagged [sums-of-squares]
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163 questions
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Asymptotic behavior of sumsets of squares with restricted congruence conditions
Recall that if $A$ and $B$ are both subsets of the integers, then $A+B=\{a+b:a \in A,b \in B\}$.
Lagrange's four-square theorem states that if $A$ is the set of squares, then $4A=A+A+A+A=\mathbb{N}$.
...
- 101
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249
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Possible solution to the 3x3 magic square of squares problem
I think I might have a solution to the problem of the existence of 3x3 magic square of squares problem and have written a ten page paper detailing most of the stuff, but I fear no-one is going to open ...
- 101
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Infinite sequence of PSD non-moments in two variables
Define a 2d sequence to be a mapping $a: \mathbb{N}^2 \to \mathbb{R}$ (where $\mathbb{N} = \{0, 1, \dots\}$). Here are two definitions of types of 2d sequences:
We say that a 2d sequence $a$ is a ...
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Which integers can be expressed as $P(t)^2 + Q(t)^2 + R(t)^5$?
Inspired by this article and that one, I have two questions:
(1) Is the question of whether every integer can be expressed in the form $x^2 + y^2 + z^5$ ($x$, $y$, $z$ in $\mathbb{Z}$) an open problem?...
- 655
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Square roots and prime numbers
Definitions:
Here I present a novel conjecture using basic mathematical tools like the sum of the
divisors of an integer $n$ called $\sigma(n)$, the sum of the squares of the positive divisors of n ...
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278
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Is there a theorem which provides conditions under which a power series satisfies the reciprocal root sum law?
Kalman - Six ways to sum a series discusses Euler's original proof for the Basel problem $\sum\limits_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6} $:
$$\frac{\sin(\sqrt x)}{\sqrt x} = 1- \frac{x}{3!}+ \...
- 541
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Decomposition in two hermitian squares over ring of integer of CM fields
Let $k$ a CM-field of conjugation $\bar \cdot$ and maximal totally real subfield $k^+$ and $k^{++}$ its positive part $k^{++}$.
Given a totally positive element $x$ in the ring of integers of $k^+$, ...
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Jacobi symbols for two-square sums of primes
Given a prime $p\equiv 1\pmod 4$, Fermat's two-squares theorem discovered by Girard
states that there exists two integers $A,B$ such that
$p=A^2+B^2$.
For all primes up to $10^7$ the integers $A$ and $...
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What is the possible reminders modulo 4 of an "odd part" of a polynomial?
Let $f(x)$ be a polynomial with integer coefficients. Let $f(x) = 2^k \cdot m$ where $m$ is odd. The questions are
What are the possible values of $m \mod 4$ (1, 3 or both)? I want the algorithm ...
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Is there an upper bound on the number of representations as a sum of squares?
I am interested in finding upper bounds for the Sum of squares function defined as $r_k(n) = |\{(a_1, a_2, \ldots, a_k) \in \mathbb{Z}^k \ : \ n = a_1^2 + a_2^2 + \cdots + a_k^2\}|$ whenever the ...
- 313
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A simple way to bound the density of sums of two odd squares
Define
$$S(x) ~=~ \# \left\{ n^2+m^2\leq x : n,m\in\mathbb{N}\right\}$$
Landau (1908) proved that with
$$ B(x) ~=~ K\,\frac{x}{ \sqrt{\log x}} ~~\text{ one has}~~~ \lim \limits_{x\to \infty} \frac{S(...
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Number of partitions of set restricted by sum of square of part size
Let $p_1^{a_1}p_2^{a_2}\cdots$ denotes the integer partition of $n$, i.e. $a_1p_1+a_2p_2+\cdots=n$. Or equivalently $m_1+m_2+\cdots=n$. It is known that the number of partitions of set $\{x_1,x_2,\...
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Representing $x^6-4$ as a sum of two squares
Prove that there exist infinitely many integers $x$ such that integer $P(x)=x^6-4$ is a sum of two squares of integers.
Equivalently, prove that $x^3-2$ and $x^3+2$ are simultaneously sums of two ...
- 7,172
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884
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Symmetric version of Hilbert's seventeenth problem?
Artin's solution to Hilbert's seventeenth problem tells us that a multivariate polynomial $f$ takes only non-negative values over the reals if and only if it is a sum of squares of rational functions.
...
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1-degree SOS proof refutes Linear Programming
I am trying to understand Sums-of-Squares proof systems.
A degree $d$ Sums-of-Squares refutation for a set of polynomial equations $P = \{p_1(x) = 0, ..., p_m(x) = 0\}$ is defined as
$\sum_{i=1}^m g_i(...
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Asymptotic analysis of a peculiar sum of squares sequence
Let $a,b$ be two positive integers. Let the sequence $\{s_n\}_n$ be the set of all possible sums of squares $a^2+b^2$, such that they are in ascending order
\begin{align*}
& n=1 & s_1=1^2+1^2=...
- 151
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240
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Bounds on largest possible square in sum of two squares
Suppose we are given integers $k,c$ such that $k=1+c^2$.
Let $n$ be an odd integer and suppose that $k^n=a_i^2+b_i^2$ for distinct positive integers $a_i<b_i$ and $i\le d$. That is, there are $d$ ...
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Sum of squares squared in an arithmetic progression
Let $r(n)$ be the number of ways to write $n$ as a sum of two squares and $(a,q)=1$.
What is known about
$$
\sum_{n \le x,n \equiv a (\text{mod} \, q)} r(n)^2 \quad?
$$
I am looking for uniform ...
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Infinite sum of Laguerre polynomials: is $\sum_{k=0}^{\infty}\frac{n!}{(k+n)!}x^kL_n^k(x)^2=e^x+P(x)$, with $P$ a polynomial of degree $2n-1$?
I have the following expression:
$$
\sum_{k=0}^{\infty}\frac{n!}{(k+n)!}x^k(L_n^k(x))^2,
$$
where
$$
L_n^k(x)=\sum_{j=0}^n(-1)^j\binom{n+k}{n-j}\frac{x^j}{j!}
$$
is the usual associated Laguerre ...
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On $w^4+x^4+y^2+z^2$ over a number field
In 1921 Siegel confirmed a conjecture of Hilbert by proving that for any number field $K$ each element of
$$K_{\geq0}=\{a\in K:\ \sigma(a)\geq0\ \mbox{for all}\ \sigma\in\mathrm{Gal}(K/\mathbb Q)\}$$ ...
- 15.6k
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How often is the value of a quadratic polynomial equal to a sum of two integer squares?
Let $b:\mathbb N\to \{0,1\}$ be the indicator function of integers that are a sum of two non-zero integer squares. Let $f(t)\in \mathbb Z[t]$ be an irreducible polynomial of degree $2$ with positive ...
- 3,062
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Representing $x^3-2$ as a sum of two squares
Prove that there exist infinitely many integers $x$ such that integer $P(x)=x^3-2$ is a sum of two squares of integers.
Ideally, I am looking for a proof method that also applies for other $P(x)$, ...
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Bounding sum of square roots in function of the sum value [closed]
Knowing the value of $S=\sum_{k=1}^n s_k$ with $s_k\geq 0$,
is it possible to obtain an upper bound on $\sum_{k=1}^n\sqrt{s_k}$ better than $n \times \sqrt{\max_{1\leq k\leq n} s_k}$ ?
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The relationship between the symmetric tensor product and sum of squares
I would like to understand deeply the relationship between the symmetric tensor product of order 2 and the sum of squares.
For me, it is clear that the symmetric tensor product or order $d$ is ...
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Gaps between combinations of squares of integers
Let $\theta$ be a positive irrational number and $S=\{\theta n^2+m^2: n, m\in \mathbb{N}\}$. The elements of $S$ can be written as a sequence of strictly increasing numbers $\{s_n\}$. My question is ...
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Sum of squares and divisibility
Consider an integer of the form $$N = 1 + \sum_{i=1}^r d_i^2$$ where $d_i \in \mathbb{N}_{\ge 3}$ and $d_i^2$ divides $N$.
Question: Must $r$ be greater than or equal to $9$?
Checking (with SageMath): ...
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Explanation of several unpublished remarks of Gauss on representations of a given number as sums of two, three and four squares
Remark 1: On p.384 of volume 3 of Gauss's Werke, which is a part of an unpublished treatise on the arithmetic geometric mean, Gauss makes the following remark:
On the theory of the division of ...
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Representing a symmetric polynomial as a conical sum of squares
This question in inspired by the recent solution to another question.
The following inequality for monomial symmetric polynomials in 4 positive variables $x_1,x_2,x_3,x_4$:
$$m_{(4, 3, 2, 1)} + m_{(4, ...
- 34.3k
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Witt ring of a field with Pythagoras number $2$
I am currently looking at a few simple properties of the Witt ring of a field $K$ (by which I mean the ring of Witt classes of quadratic forms, not the ring of Witt vectors), which are clearly true ...
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Linear independence of complex polynomials and a "sum of squares" conjecture
This will take me some time to explain. Let $n \geq 2$ be a fixed integer. Let $p_i(z)$, for $i = 1,\ldots,n$ be $n$ nonzero complex polynomials of degree at most $n-1$. I am interested in ...
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Representation of a power of a quadratic form
Let $A=A^t$ be a non-singular symmetric matrix. For any multi-index $\gamma=(\gamma_1,\dots,\gamma_n)$ of degree $\vert \gamma\vert=\gamma_1+\dots+\gamma_n=2d$, let $b_\gamma=\frac{\partial}{\partial ...
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Does there always exist a(n uniform) polynomial that makes a positive definite symmetric matrix with polynomial entries into a sum of squares?
Suppose that I have a square and positive definite for every evaluation $x\in\mathbb{R}^{n}$ symmetric matrix $M(x)\in(\mathbb{R}[x])^{s\times s}.$
Does there always exist a polynomial $p(x)\in\...
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Lower bounds on lengths of sum-of-squares representations of particular polynomials
I am looking for literature on the problem of finding minimal (in the sense of number of terms) sum-of-squares representations of particular non-negative multivariate polynomials with rational ...
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Can I prove that a polynomial representing the 4th moment of a weighted-sum of random variables is a sos?
I am looking at the 4th central moment of a weighted-sum of correlated random variables, which takes the form
$$\mu_4 = \sum_{i,j,k,l=1}^n w_i w_j w_k w_l \mu_{ijkl}$$
where $\mu_{ijkl}$ are the ...
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A generalization of partition function to the sums of squares
The well known partition function $p(n)$ is defined as the number of ways to represent $n$ as the sum of natural numbers. An asymptotic formula for $p(n)$ is
$$p(n)\sim\frac{1}{4n\sqrt{3}}\exp\left(\...
user167505
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Is there yet an example of a non-negative convex polynomial that cannot be written as a sum-of-squares?
I have read that it remains an open question, whether an example can be constructed of a non-negative convex polynomial that cannot be written as a sum-of-squares. My reading includes the following ...
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Is it true that $\{x^4+y^2+z^2:\ x,y,z\in\mathbb Z[i]\}=\{a+2bi:\ a,b\in\mathbb Z\}$?
Recall that the ring of Gaussian integers is
$$\mathbb Z[i]=\{a+bi:\ a,b\in\mathbb Z\}.$$
Clearly
$$(a+bi)^2=a^2-b^2+2abi\ \ \mbox{and}\ \ (a+bi)^4=(a^2-b^2)^2-4a^2b^2+4ab(a^2-b^2)i.$$
Question. Is it ...
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Questions on $x^2+y^2+z^2$, $x^2+y^2+2z^2$ and $x^2+2y^2+3z^2$
Question 1. My computation in 2018 suggests that a sum of two integer squares is a sum of three nonzero integer squares if and only if it is not of the form
$$4^km\ \ (k=0,1,2,\ldots;\ \ m=1,2,5,10,13,...
- 15.6k
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univariate integer version of Hilbert's 17th problem
Let $f(x)$ be a polynomial of degree $d$ with integer coefficients such that $f(x)\geqslant 0$ for all real $x$. Is it necessarily true that there exists an integer $N(d)$ such that $N(d)\cdot f$ is a ...
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$7n=x^2+2y^2+4z^2$ with or without $x^2\equiv y^2\equiv z^2\pmod 7$
It is well known that any positive odd integer can be written as $x^2+2y^2+4z^2$ with $x,y,z\in\mathbb Z$.
Question 1. Whether for any odd integer $n>93$ there are $x,y,z\in\mathbb Z$ such that $7n=...
- 15.6k
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1
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$x^2+7y^2=2^n$ and sums of four squares
Lagrange's four square theorem states that each $m\in\mathbb N=\{0,1,2,\ldots\}$ can be written as a sum of four squares.
Recently, I found that the diophantine equation $x^2+7y^2=2^n$ has certain ...
- 15.6k
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SOS polynomials with rational coefficients
Suppose we are given a univariate polynomial with rational coefficients, $p \in \Bbb Q [x]$, and are told that $p$ can be expressed as the sum of $k$ squares of polynomials with rational coefficients. ...
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On a sum of squares representation
We know $p a^2+q b^2+r ab$ can be represented as square (trivially) when $$p,q\geq0$$
$$r^2=|4pq|$$
holds and as a sum of squares (again trivially) of form $(m a+n b)^2$ under readily explainable ...
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The power of chi-square test
Under the null hypothesis, if we have
$$\sqrt{n} \vec{x} \, \rightarrow_d \, N(0, I_p),$$
the test statistic can be construct as:
$$\hat{\Psi} = n \vec{x}^{\top} \vec{x} \, \rightarrow_d \,\chi^2_p.$$
...
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Find the integer part of the sum [closed]
Find the integer part of this
sum
Right answer is 200000000010000000000. But i don’t know how to solve it.
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What is the relation between different generalizations of linear programming?
Linear programming subsumed by each of
Semidefinite programming (SDP)
Convex programming (CXP)
SOS programming (SSP)
Is there any relation between each pair in the three?
Are all three equivalent in ...
- 1,826
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Sum of squares of polynomials in one variable with missing powers
As we known, a positive polynomial in $\mathbb{R}\left[x\right]$ can be expressed as a sum of squares of polynomials.
The problem is that whether this holds if some powers is missed.
Let $A$ be a ...
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Matrix whose entries are given by polynomial $A_{ij} = p(\lambda_i, \lambda_j)$; when is it positive semidefinite?
Let $A$ be a matrix whose entries are given by a polynomial,
$$
A_{ij} = p(\lambda_i, \lambda_j)
$$
where $p(\lambda_i,\lambda_j) = p(\lambda_j,\lambda_i)$ is symmetric.
Are there standard methods ...
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Small linear relations between primitive Pythagorean triples $\mathsf I$
Say $a^2+b^2=c^2$ is a primitive Pythagorean triple. Then consider the Linear Diophantine Equation
$$ua^2+vb^2+xab+ybc+zca=0$$ where $(u,v,x, y, z)\in\mathbb Z^4$ are variables. If $(u,v,x, y, z)\neq(...
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Efficient computation of $\sum_{i=1}^{\sqrt{n}} i^2\cdot\left\lfloor{\frac n{i^2}}\right\rfloor$
I need to compute efficiently the sum
$$
\sum_{i=1}^{\sqrt{n}} i^2\cdot\left\lfloor{\frac n{i^2}}\right\rfloor.
$$
We can do this in $O({\sqrt{n}})$ but I need a faster algorithm: for example, it ...
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