Questions tagged [sums-of-squares]

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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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0 votes
0 answers
127 views

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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4 votes
3 answers
212 views

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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5 votes
0 answers
272 views

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 $$K_{\geq0}=\{a\in K:\ \sigma(a)\geq0\ \mbox{for all}\ \sigma\in\mathrm{Gal}(K/\mathbb Q)\}$$ ...
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4 votes
1 answer
193 views

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 ...
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15 votes
1 answer
794 views

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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-3 votes
1 answer
272 views

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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0 votes
1 answer
97 views

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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2 votes
2 answers
271 views

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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25 votes
3 answers
2k views

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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10 votes
1 answer
356 views

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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6 votes
1 answer
246 views

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, ...
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4 votes
1 answer
115 views

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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0 votes
1 answer
121 views

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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3 votes
0 answers
114 views

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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2 votes
0 answers
38 views

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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  • 522
1 vote
0 answers
54 views

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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1 vote
1 answer
150 views

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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  • 173
7 votes
2 answers
745 views

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(\...
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11 votes
0 answers
317 views

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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  • 173
18 votes
1 answer
595 views

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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-1 votes
1 answer
233 views

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,...
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19 votes
0 answers
484 views

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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-1 votes
1 answer
176 views

$7n=x^2+2y^2+4z^2$ 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=...
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2 votes
1 answer
484 views

$x^2+7y^2=2^n$ 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 ...
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10 votes
1 answer
897 views

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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  • 1,643
0 votes
1 answer
101 views

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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4 votes
1 answer
272 views

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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1 vote
0 answers
82 views

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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1 vote
0 answers
75 views

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 ...
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3 votes
0 answers
74 views

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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  • 29
0 votes
1 answer
130 views

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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1 vote
1 answer
155 views

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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3 votes
1 answer
434 views

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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0 votes
1 answer
190 views

Write $n^2$ as $x^2+y^2+2\times4^z$ or $x^2+y^2+5\times 4^z$

In March 2018, I formulated the following somewhat curious question. Question 1. Whether for any integer $n>1$ there is a nonnegative integer $k$ such that $n^2-2\times 4^k$ or $n^2-5\times 4^k$ ...
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3 votes
1 answer
283 views

How are natural numbers that cannot be written as a sum of exactly four squares of naturals characterized? [closed]

This is most probably asked here already in some other form (as it seems to be a very basic question) but since I have't found some question very similar to this one I feel obliged to ask (also ...
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9 votes
1 answer
290 views

Hahn's approach to Hilbert's 17th problem?

The Wikipedia article on Hahn Series mentions mentioned that these were studied by Hahn "in his approach to Hilbert's seventeenth problem". Is this correct? If so, what was this approach, ...
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  • 4,184
1 vote
0 answers
71 views

Are those two Sum-Of-Squares approach for unconstrained polynomial optimization related?

I found 2 approaches to solve an unconstrained polynomial optimization problem using the Lasserre / SOS hierarchy: $$ \inf_{x\in\mathbb{R}^n}\quad p(x), $$ where $p$ is a polynomial of even degree ...
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  • 547
2 votes
0 answers
175 views

Sums of squares in fields

Which fields $k$ have the property that any sum of squares is a square ? Are there elegant characterizations and/or classifications known for this type of field ? And what if we replace "fields" by "...
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  • 3,271
1 vote
0 answers
93 views

Lagrange's four squares theorem in other fields [duplicate]

Is something known about analogues of Lagrange's four squares theorem in number fields other than $\mathbb{Q}$? I'm more interested in the case of finite extensions of $\mathbb{Q}$. For example, is ...
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6 votes
1 answer
295 views

Spherical Bessel functions. Sum of squares

In (1) there is a property of spherical Bessel functions, which's derivation I can not find in the literature. ${\mathsf{j}_{n}^{2}}\left(z\right)+{\mathsf{y}_{n}^{2}}\left(z\right)=\sum_{k=% 0}^{n}\...
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4 votes
0 answers
480 views

Four-square Conjecture

Lagrange's four-square theorem states that every nonnegative integer can be written as the sum of four squares. My following conjecture is much stronger than this classical theorem. Four-square ...
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-1 votes
1 answer
356 views

Positive integers written as $\frac{a(a+1)}2+\frac{b(b+1)}2+4^c5^d$

Let $\mathbb N=\{0,1,2,\ldots\}$. Those $T_n:=n(n+1)/2$ with $n\in\mathbb N$ are called triangular numbers. It is well known that $$\{T_a+T_b+T_c:\ a,b,c\in\mathbb N\}=\mathbb N\tag{1}$$ which was ...
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1 vote
1 answer
148 views

Constrained optimization of sum of squares polynomials

Consider the problem $$ \min p(x) \text{ subject to } g_j(x)\le 0 \quad p,g_j\in\text{SOS}, \qquad (*) $$ i.e. $p,g_j$ ($j=1,\ldots,m$) are sum of squares (SOS) polynomials. Can this problem be ...
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2 votes
0 answers
39 views

Sums of squares lengths in coordinates rings of plane curves

let $C\subseteq\mathbb{P}^2$ be a plane algebraic curve and let $\mathbb{R}[C]$ be its real coordinate ring. Let $d\geq 1$ and let $p(d)$ be the smallest number such that every sum of squares in $\...
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  • 51
18 votes
1 answer
712 views

Lagrange four-squares theorem --- deterministic complexity

Lagrange's four-squares theorem states that every natural number can be represented as the sum of four integer squares. Rabin and Shallit gave a randomised algorithm that finds one of these solutions ...
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  • 283
9 votes
1 answer
850 views

Sums of two squares in arithmetic progressions

Let $r(n)$ denote the number of representations of $n$ as the sum of two squares. Are there any known results on $$\sum _{n\leq x\atop {n\equiv a(q)}}r(n)$$ and in particular is there an asymptotic ...
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0 votes
0 answers
114 views

Does positivstellensatz and SOS proof system help here?

I have a system of $m$ homogeneous degree $2$ polynomial equations in $\mathbb Z[x_1,\dots,x_n]$ where $m=poly(n)$. Take $$f_1(x_1,\dots,x_n)=0$$ $$\dots$$ $$f_m(x_1,\dots,x_n)=0$$ to be the system. ...
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  • 1,728
0 votes
0 answers
118 views

Do many homogeneous polynomials help in faster integer root extraction?

Given $n$ homogeneous algebraically independent total degree $2$ polynomials with no $x_1^2,\dots,x_n^2$ variable in $\mathbb Z[x_1,\dots,x_n]$ with promise that it has non-zero integer roots with ...
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  • 1,728
3 votes
0 answers
170 views

On sums of three squares

Let $\mathbb N=\{0,1,2,\ldots\}$. By the Gauss-Legendre theorem on sums of three squares, any $n\in\mathbb N$ with $n\equiv1,2\pmod4$ can be written as $x^2+y^2+z^2$ with $x,y,z\in\mathbb N$. Clearly, ...
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