# Questions tagged [sums-of-squares]

The sums-of-squares tag has no usage guidance.

135
questions

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votes

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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 ...

**0**

votes

**1**answer

103 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 ...

**2**

votes

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83 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 ...

**2**

votes

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34 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\...

**1**

vote

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43 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 ...

**0**

votes

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93 views

### Is there a characterization of Pythagorean ellipses?

A quadratic bivariate
$$\Phi(x,y)=a_{2,0}x^2+a_{1,1}xy+a_{0,2}y^2+a_{1,0}x+a_{0,1}y+a_{0,0}=0$$
describes an ellipse if the constraint
$$a_{1,1}^2-4a_{2,0}a_{0,2}<0$$
holds.
Given an ellipse $\Phi(...

**1**

vote

**1**answer

133 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 ...

**7**

votes

**2**answers

569 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(\...

**11**

votes

**0**answers

304 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 ...

**16**

votes

**1**answer

460 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 ...

**-1**

votes

**1**answer

213 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,...

**20**

votes

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447 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 ...

**-1**

votes

**1**answer

168 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=...

**2**

votes

**1**answer

449 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 ...

**11**

votes

**1**answer

840 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. ...

**0**

votes

**1**answer

94 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 ...

**4**

votes

**1**answer

182 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:
$$\hat{\Psi} = n \vec{x}^{\top} \vec{x} \, \rightarrow_d \,\chi^2_p.$$
...

**1**

vote

**0**answers

78 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.

**1**

vote

**0**answers

74 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 ...

**3**

votes

**0**answers

67 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 ...

**0**

votes

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55 views

### Sum of four squares from sum of $k\geq5$ squares - scaled version $\mathsf I$

Every natural number is sum of $4$ squares and every natural number bigger than $169$ is sum of five squares.
Suppose $n>169$ and $x'x=n$ at a vector $x\in\mathbb Z^5$ then can we find a matrix $M\...

**0**

votes

**1**answer

122 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 ...

**1**

vote

**1**answer

149 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(...

**3**

votes

**1**answer

324 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 ...

**0**

votes

**1**answer

182 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 $n^2-5\times 4^k$ ...

**3**

votes

**1**answer

213 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 ...

**6**

votes

**0**answers

149 views

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

The Wikipedia article on Hahn Series mentions that these were studied by Hahn "in his approach to Hilbert's seventeenth problem".
Is this correct? If so, what was this approach, and where can I ...

**1**

vote

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65 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 ...

**2**

votes

**0**answers

144 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 "...

**1**

vote

**0**answers

88 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 ...

**6**

votes

**1**answer

200 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}\...

**4**

votes

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462 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 ...

**-1**

votes

**1**answer

342 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 ...

**1**

vote

**1**answer

115 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 ...

**2**

votes

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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 $\...

**18**

votes

**1**answer

662 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 ...

**9**

votes

**1**answer

739 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 ...

**0**

votes

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93 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.
...

**0**

votes

**0**answers

114 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 ...

**3**

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164 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, ...

**7**

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271 views

### Is 100 the only Leyland number that is a square?

Leyland numbers (named for Paul Leyland) are positive integers of the form $x^y + y^x$ , where x and y are naturals > 1, and also the number 3. The OEIS link is https://oeis.org/A076980
I thought ...

**6**

votes

**2**answers

1k views

### Sums of squares of primes [closed]

Question: What is the least number that is a sum of three squares of primes in exactly six ways?
... I know it is not research mathematics. Happy new year!
EDIT: Now that it is answered I should ...

**2**

votes

**1**answer

165 views

### Sieve bound for the sum of two squares

Let $$S(n) = \sum_{p \le n} b(n-p),$$ where
$b(a)=1$ is $a$ is a sum of two squares of positive integers and $b(a)=0$ otherwise.
Trivially by PNT we have
$$S(n) \ll \sum_{p \le n}1 \le \frac{n}{\log n}...

**5**

votes

**0**answers

218 views

### Can we write each positive integer as $x^2+y^2+\varphi(z^2)$?

As odd squares are congruent to $1$ modulo $8$, any integer of the form $4^k(8m+7)$ with $k,m\in\mathbb N=\{0,1,2,\ldots\}$ cannot be written as the sum of three squares.
To avoid such congruence ...

**6**

votes

**0**answers

269 views

### Legendre's three-square theorem and squared norm of integer matrices

Let $\mathbb{N}$ be the set of non-negative integers. Let $E_n$ be the set of integers which are the sum of $n$ squares. Let $F_n$ be the set of integers of the form $\Vert A \Vert^2$ with $A \in M_n(...

**11**

votes

**2**answers

379 views

### Extension of Dickson's theorem on integers of the form $a^2+b^2+2c^2$

Theorems V in this paper of L.E. Dickson states that the following two sets are equal. $$E=\{a^2+b^2+2c^2 \ | \ a,b,c \in \mathbb{Z}\} \ \text{ and } \ F=\mathbb{N} \setminus \{4^k(16n+14) \ | \ k,n \...

**35**

votes

**3**answers

2k views

### Lagrange four squares theorem

Lagrange's four square theorem states that every non-negative integer is a sum of squares of four non-negative integers. Suppose $X$ is a subset of non-negative integers with the same property, that ...

**7**

votes

**1**answer

385 views

### Can a positive polynomial on sphere be represented as the sum of squares of spherical harmonics

Let $p\in {\mathbb{R}}[x_1,\ldots, x_d]$ be a homogenous polynomial degree $2n$. We know that if $p$ is positive on $[-\pi,\pi]^d$, $p$ is sum of squares polynomial, i.e. $p$ can be witten as sum of ...

**2**

votes

**2**answers

360 views

### Number of ways to write an integer as a sum of squares modulo $k$

Given a natural number $n$ and an element $k \in \mathbb{Z}_n$, how many solutions are there in $\mathbb{Z}_n$ to the equation $x^2+y^2 =k$? That is, I'm wondering whether there is a mod-$n$ version ...

**13**

votes

**1**answer

728 views

### For a proof of the three-square theorem without using Dirichlet's theorem on primes in arithmetic progressions

The three-square theorem states that $n\in\mathbb N=\{0,1,2,\ldots\}$ is the sum of three squares if and only if it is not of the form $4^k(8m+7)$ ($k,m\in\mathbb N$). This was first proved by ...