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5 votes
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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!}+ \...
pie's user avatar
  • 421
0 votes
0 answers
33 views

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 $k^+$, ...
user70925's user avatar
  • 313
11 votes
2 answers
567 views

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 $...
Roland Bacher's user avatar
1 vote
0 answers
132 views

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 ...
Denis Shatrov's user avatar
5 votes
1 answer
478 views

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 ...
MathqA's user avatar
  • 313
6 votes
1 answer
357 views

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(...
Karl Fabian's user avatar
  • 1,571
0 votes
0 answers
36 views

Finding the minimum representation of a positive integer as the sum of three nonzero squares

[n.b. This is a crosspost/nuancing of this MSE post.] Assume we have a positive integer $n$ which is already known to be the sum of the squares of three nonzero integers — for example, $$ n = 451 = 19^...
Kieren MacMillan's user avatar
3 votes
0 answers
217 views

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,\...
tony's user avatar
  • 405
8 votes
1 answer
614 views

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 ...
Bogdan Grechuk's user avatar
12 votes
3 answers
852 views

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. ...
Timothy Chow's user avatar
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0 votes
0 answers
82 views

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(...
Tom Keaton's user avatar
2 votes
1 answer
148 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=...
TheVal's user avatar
  • 151
2 votes
1 answer
235 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 $d$ ...
TheSimpliFire's user avatar
0 votes
0 answers
191 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 ...
toshi's user avatar
  • 130
4 votes
3 answers
404 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 ...
L. Proz's user avatar
  • 83
5 votes
0 answers
284 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 $$K_{\geq0}=\{a\in K:\ \sigma(a)\geq0\ \mbox{for all}\ \sigma\in\mathrm{Gal}(K/\mathbb Q)\}$$ ...
Zhi-Wei Sun's user avatar
  • 14.6k
5 votes
1 answer
304 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 ...
Dr. Pi's user avatar
  • 3,009
15 votes
2 answers
1k 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 $P(x)$, ...
Bogdan Grechuk's user avatar
-3 votes
1 answer
2k 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 $n \times \sqrt{\max_{1\leq k\leq n} s_k}$ ?
OmarR's user avatar
  • 67
0 votes
1 answer
234 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 ...
Tio Miserias's user avatar
2 votes
2 answers
280 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 ...
J. J.'s user avatar
  • 21
26 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): ...
Sebastien Palcoux's user avatar
11 votes
1 answer
476 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 ...
user2554's user avatar
  • 1,969
6 votes
1 answer
294 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, ...
Max Alekseyev's user avatar
4 votes
1 answer
143 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 ...
Captain Lama's user avatar
0 votes
1 answer
135 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 ...
Malkoun's user avatar
  • 5,118
3 votes
0 answers
150 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 ...
Khazhgali Kozhasov's user avatar
2 votes
0 answers
54 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\...
Hvjurthuk's user avatar
  • 573
1 vote
0 answers
66 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 ...
Alex Elzenaar's user avatar
1 vote
1 answer
257 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 ...
Brian's user avatar
  • 173
8 votes
2 answers
1k 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(\...
user avatar
11 votes
0 answers
358 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 ...
Brian's user avatar
  • 173
18 votes
1 answer
668 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 ...
Zhi-Wei Sun's user avatar
  • 14.6k
-1 votes
1 answer
250 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,...
Zhi-Wei Sun's user avatar
  • 14.6k
19 votes
0 answers
521 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 ...
Fedor Petrov's user avatar
-1 votes
1 answer
182 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=...
Zhi-Wei Sun's user avatar
  • 14.6k
2 votes
1 answer
536 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 ...
Zhi-Wei Sun's user avatar
  • 14.6k
10 votes
1 answer
1k 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. ...
Gautam's user avatar
  • 1,703
0 votes
1 answer
114 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 ...
VS.'s user avatar
  • 1,826
4 votes
1 answer
314 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: $$\hat{\Psi} = n \vec{x}^{\top} \vec{x} \, \rightarrow_d \,\chi^2_p.$$ ...
香结丁's user avatar
  • 331
1 vote
0 answers
106 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.
Sergey's user avatar
  • 11
1 vote
0 answers
81 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 ...
VS.'s user avatar
  • 1,826
3 votes
0 answers
81 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 ...
Chivul's user avatar
  • 129
0 votes
1 answer
147 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 ...
Felix Huber's user avatar
1 vote
1 answer
170 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(...
VS.'s user avatar
  • 1,826
4 votes
1 answer
714 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 ...
prolific's user avatar
0 votes
1 answer
196 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 $n^2-5\times 4^k$ ...
Zhi-Wei Sun's user avatar
  • 14.6k
3 votes
1 answer
379 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 ...
user avatar
9 votes
1 answer
324 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, ...
Tobias Fritz's user avatar
  • 6,177
1 vote
0 answers
99 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 ...
guigux's user avatar
  • 607