Questions tagged [sums-of-squares]
The sums-of-squares tag has no usage guidance.
149
questions
10
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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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0
votes
0
answers
73
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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(...
2
votes
1
answer
90
views
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
2
votes
1
answer
218
views
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 $d$ ...
- 421
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0
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161
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 ...
- 120
4
votes
3
answers
288
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 ...
- 83
5
votes
0
answers
280
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
$$K_{\geq0}=\{a\in K:\ \sigma(a)\geq0\ \mbox{for all}\ \sigma\in\mathrm{Gal}(K/\mathbb Q)\}$$ ...
- 13.7k
4
votes
1
answer
237
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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 ...
- 2,862
16
votes
1
answer
922
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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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-3
votes
1
answer
1k
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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 $n \times \sqrt{\max_{1\leq k\leq n} s_k}$ ?
- 57
0
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1
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145
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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 ...
- 139
2
votes
2
answers
277
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 ...
- 21
25
votes
3
answers
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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): ...
- 24.7k
11
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1
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447
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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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6
votes
1
answer
279
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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, ...
- 28.7k
4
votes
1
answer
121
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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 ...
- 252
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1
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127
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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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votes
0
answers
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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 ...
2
votes
0
answers
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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\...
- 561
1
vote
0
answers
61
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 ...
1
vote
1
answer
203
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}$$
where $\mu_{ijkl}$ are the ...
- 173
7
votes
2
answers
887
views
A generalization of partition function to the sums of squares
The well known partition function $p(n)$ is defined as the number of ways to represent $n$ as the sum of natural numbers. An asymptotic formula for $p(n)$ is
$$p(n)\sim\frac{1}{4n\sqrt{3}}\exp\left(\...
user167505
11
votes
0
answers
346
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 ...
- 173
18
votes
1
answer
639
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
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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
$$4^km\ \ (k=0,1,2,\ldots;\ \ m=1,2,5,10,13,...
- 13.7k
19
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0
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501
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univariate integer version of Hilbert's 17th problem
Let $f(x)$ be a polynomial of degree $d$ with integer coefficients such that $f(x)\geqslant 0$ for all real $x$. Is it necessarily true that there exists an integer $N(d)$ such that $N(d)\cdot f$ is a ...
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-1
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1
answer
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$7n=x^2+2y^2+4z^2$ with or without $x^2\equiv y^2\equiv z^2\pmod 7$
It is well known that any positive odd integer can be written as $x^2+2y^2+4z^2$ with $x,y,z\in\mathbb Z$.
Question 1. Whether for any odd integer $n>93$ there are $x,y,z\in\mathbb Z$ such that $7n=...
- 13.7k
2
votes
1
answer
509
views
$x^2+7y^2=2^n$ and sums of four squares
Lagrange's four square theorem states that each $m\in\mathbb N=\{0,1,2,\ldots\}$ can be written as a sum of four squares.
Recently, I found that the diophantine equation $x^2+7y^2=2^n$ has certain ...
- 13.7k
10
votes
1
answer
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views
SOS polynomials with rational coefficients
Suppose we are given a univariate polynomial with rational coefficients, $p \in \Bbb Q [x]$, and are told that $p$ can be expressed as the sum of $k$ squares of polynomials with rational coefficients. ...
- 1,683
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1
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102
views
On a sum of squares representation
We know $p a^2+q b^2+r ab$ can be represented as square (trivially) when $$p,q\geq0$$
$$r^2=|4pq|$$
holds and as a sum of squares (again trivially) of form $(m a+n b)^2$ under readily explainable ...
- 1,776
4
votes
1
answer
297
views
The power of chi-square test
Under the null hypothesis, if we have
$$\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.$$
...
- 319
1
vote
0
answers
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Find the integer part of the sum [closed]
Find the integer part of this
sum
Right answer is 200000000010000000000. But i don’t know how to solve it.
- 11
1
vote
0
answers
79
views
What is the relation between different generalizations of linear programming?
Linear programming subsumed by each of
Semidefinite programming (SDP)
Convex programming (CXP)
SOS programming (SSP)
Is there any relation between each pair in the three?
Are all three equivalent in ...
- 1,776
3
votes
0
answers
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Sum of squares of polynomials in one variable with missing powers
As we known, a positive polynomial in $\mathbb{R}\left[x\right]$ can be expressed as a sum of squares of polynomials.
The problem is that whether this holds if some powers is missed.
Let $A$ be a ...
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votes
1
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134
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Matrix whose entries are given by polynomial $A_{ij} = p(\lambda_i, \lambda_j)$; when is it positive semidefinite?
Let $A$ be a matrix whose entries are given by a polynomial,
$$
A_{ij} = p(\lambda_i, \lambda_j)
$$
where $p(\lambda_i,\lambda_j) = p(\lambda_j,\lambda_i)$ is symmetric.
Are there standard methods ...
- 219
1
vote
1
answer
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Small linear relations between primitive Pythagorean triples $\mathsf I$
Say $a^2+b^2=c^2$ is a primitive Pythagorean triple. Then consider the Linear Diophantine Equation
$$ua^2+vb^2+xab+ybc+zca=0$$ where $(u,v,x, y, z)\in\mathbb Z^4$ are variables. If $(u,v,x, y, z)\neq(...
- 1,776
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votes
1
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Efficient computation of $\sum_{i=1}^{\sqrt{n}} i^2\cdot\left\lfloor{\frac n{i^2}}\right\rfloor$
I need to compute efficiently the sum
$$
\sum_{i=1}^{\sqrt{n}} i^2\cdot\left\lfloor{\frac n{i^2}}\right\rfloor.
$$
We can do this in $O({\sqrt{n}})$ but I need a faster algorithm: for example, it ...
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votes
1
answer
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Write $n^2$ as $x^2+y^2+2\times4^z$ or $x^2+y^2+5\times 4^z$
In March 2018, I formulated the following somewhat curious question.
Question 1. Whether for any integer $n>1$ there is a nonnegative integer $k$ such that $n^2-2\times 4^k$ or $n^2-5\times 4^k$ ...
- 13.7k
3
votes
1
answer
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How are natural numbers that cannot be written as a sum of exactly four squares of naturals characterized? [closed]
This is most probably asked here already in some other form (as it seems to be a very basic question) but since I have't found some question very similar to this one I feel obliged to ask (also ...
user153451
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votes
1
answer
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Hahn's approach to Hilbert's 17th problem?
The Wikipedia article on Hahn Series mentions mentioned that these were studied by Hahn "in his approach to Hilbert's seventeenth problem".
Is this correct? If so, what was this approach, ...
- 5,128
1
vote
0
answers
85
views
Are those two Sum-Of-Squares approach for unconstrained polynomial optimization related?
I found 2 approaches to solve an unconstrained polynomial optimization problem using the Lasserre / SOS hierarchy:
$$
\inf_{x\in\mathbb{R}^n}\quad p(x),
$$
where $p$ is a polynomial of even degree ...
- 607
2
votes
0
answers
188
views
Sums of squares in fields
Which fields $k$ have the property that any sum of squares is a square ?
Are there elegant characterizations and/or classifications known for this type of field ?
And what if we replace "fields" by "...
- 3,871
1
vote
0
answers
94
views
Lagrange's four squares theorem in other fields [duplicate]
Is something known about analogues of Lagrange's four squares theorem in number fields other than $\mathbb{Q}$?
I'm more interested in the case of finite extensions of $\mathbb{Q}$.
For example, is ...
6
votes
1
answer
338
views
Spherical Bessel functions. Sum of squares
In (1) there is a property of spherical Bessel functions, which's derivation I can not find in the literature.
${\mathsf{j}_{n}^{2}}\left(z\right)+{\mathsf{y}_{n}^{2}}\left(z\right)=\sum_{k=%
0}^{n}\...
- 65
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votes
0
answers
488
views
Four-square Conjecture
Lagrange's four-square theorem states that every nonnegative integer
can be written as the sum of four squares. My following conjecture is much stronger than this classical theorem.
Four-square ...
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-1
votes
1
answer
362
views
Positive integers written as $\frac{a(a+1)}2+\frac{b(b+1)}2+4^c5^d$
Let $\mathbb N=\{0,1,2,\ldots\}$. Those
$T_n:=n(n+1)/2$ with $n\in\mathbb N$ are called triangular numbers. It is well known that
$$\{T_a+T_b+T_c:\ a,b,c\in\mathbb N\}=\mathbb N\tag{1}$$
which was ...
- 13.7k
1
vote
1
answer
184
views
Constrained optimization of sum of squares polynomials
Consider the problem
$$
\min p(x) \text{ subject to } g_j(x)\le 0
\quad
p,g_j\in\text{SOS},
\qquad
(*)
$$
i.e. $p,g_j$ ($j=1,\ldots,m$) are sum of squares (SOS) polynomials. Can this problem be ...
- 660
2
votes
0
answers
39
views
Sums of squares lengths in coordinates rings of plane curves
let $C\subseteq\mathbb{P}^2$ be a plane algebraic curve and let $\mathbb{R}[C]$ be its real coordinate ring.
Let $d\geq 1$ and let $p(d)$ be the smallest number such that every sum of squares in $\...
- 51
18
votes
1
answer
774
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Lagrange four-squares theorem --- deterministic complexity
Lagrange's four-squares theorem states that every natural number can be represented as the sum of four integer squares. Rabin and Shallit gave a randomised algorithm that finds one of these solutions ...
- 283
9
votes
1
answer
928
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Sums of two squares in arithmetic progressions
Let $r(n)$ denote the number of representations of $n$ as the sum of two squares. Are there any known results on $$\sum _{n\leq x\atop {n\equiv a(q)}}r(n)$$ and in particular is there an asymptotic ...
- 143