# 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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### 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 $$K_{\geq0}=\{a\in K:\ \sigma(a)\geq0\ \mbox{for all}\ \sigma\in\mathrm{Gal}(K/\mathbb Q)\}$$ ...
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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 ...
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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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This question in inspired by the recent solution to another question. The following inequality for monomial symmetric polynomials in 4 positive variables $x_1,x_2,x_3,x_4$: $$m_{(4, 3, 2, 1)} + m_{(4, ... 4 votes 1 answer 115 views ### Witt ring of a field with Pythagoras number 2 I am currently looking at a few simple properties of the Witt ring of a field K (by which I mean the ring of Witt classes of quadratic forms, not the ring of Witt vectors), which are clearly true ... 0 votes 1 answer 121 views ### Linear independence of complex polynomials and a "sum of squares" conjecture This will take me some time to explain. Let n \geq 2 be a fixed integer. Let p_i(z), for i = 1,\ldots,n be n nonzero complex polynomials of degree at most n-1. I am interested in ... 3 votes 0 answers 114 views ### Representation of a power of a quadratic form Let A=A^t be a non-singular symmetric matrix. For any multi-index \gamma=(\gamma_1,\dots,\gamma_n) of degree \vert \gamma\vert=\gamma_1+\dots+\gamma_n=2d, let b_\gamma=\frac{\partial}{\partial ... 2 votes 0 answers 38 views ### Does there always exist a(n uniform) polynomial that makes a positive definite symmetric matrix with polynomial entries into a sum of squares? Suppose that I have a square and positive definite for every evaluation x\in\mathbb{R}^{n} symmetric matrix M(x)\in(\mathbb{R}[x])^{s\times s}. Does there always exist a polynomial p(x)\in\... 1 vote 0 answers 54 views ### Lower bounds on lengths of sum-of-squares representations of particular polynomials I am looking for literature on the problem of finding minimal (in the sense of number of terms) sum-of-squares representations of particular non-negative multivariate polynomials with rational ... 1 vote 1 answer 150 views ### Can I prove that a polynomial representing the 4th moment of a weighted-sum of random variables is a sos? I am looking at the 4th central moment of a weighted-sum of correlated random variables, which takes the form$$\mu_4 = \sum_{i,j,k,l=1}^n w_i w_j w_k w_l \mu_{ijkl}$$where \mu_{ijkl} are the ... 7 votes 2 answers 745 views ### A generalization of partition function to the sums of squares The well known partition function p(n) is defined as the number of ways to represent n as the sum of natural numbers. An asymptotic formula for p(n) is$$p(n)\sim\frac{1}{4n\sqrt{3}}\exp\left(\... 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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### 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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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 $$4^km\ \ (k=0,1,2,\ldots;\ \ m=1,2,5,10,13,... 19 votes 0 answers 484 views ### univariate integer version of Hilbert's 17th problem Let f(x) be a polynomial of degree d with integer coefficients such that f(x)\geqslant 0 for all real x. Is it necessarily true that there exists an integer N(d) such that N(d)\cdot f is a ... -1 votes 1 answer 176 views ### 7n=x^2+2y^2+4z^2 with or without x^2\equiv y^2\equiv z^2\pmod 7 It is well known that any positive odd integer can be written as x^2+2y^2+4z^2 with x,y,z\in\mathbb Z. Question 1. Whether for any odd integer n>93 there are x,y,z\in\mathbb Z such that 7n=... 2 votes 1 answer 484 views ### x^2+7y^2=2^n and sums of four squares Lagrange's four square theorem states that each m\in\mathbb N=\{0,1,2,\ldots\} can be written as a sum of four squares. Recently, I found that the diophantine equation x^2+7y^2=2^n has certain ... 10 votes 1 answer 897 views ### SOS polynomials with rational coefficients Suppose we are given a univariate polynomial with rational coefficients, p \in \Bbb Q [x], and are told that p can be expressed as the sum of k squares of polynomials with rational coefficients. ... 0 votes 1 answer 101 views ### On a sum of squares representation We know p a^2+q b^2+r ab can be represented as square (trivially) when$$p,q\geq0r^2=|4pq|$$holds and as a sum of squares (again trivially) of form (m a+n b)^2 under readily explainable ... 4 votes 1 answer 272 views ### The power of chi-square test Under the null hypothesis, if we have$$\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.$$... 1 vote 0 answers 82 views ### Find the integer part of the sum [closed] Find the integer part of this sum Right answer is 200000000010000000000. But i don’t know how to solve it. 1 vote 0 answers 75 views ### What is the relation between different generalizations of linear programming? Linear programming subsumed by each of Semidefinite programming (SDP) Convex programming (CXP) SOS programming (SSP) Is there any relation between each pair in the three? Are all three equivalent in ... 3 votes 0 answers 74 views ### Sum of squares of polynomials in one variable with missing powers As we known, a positive polynomial in \mathbb{R}\left[x\right] can be expressed as a sum of squares of polynomials. The problem is that whether this holds if some powers is missed. Let A be a ... 0 votes 1 answer 130 views ### Matrix whose entries are given by polynomial A_{ij} = p(\lambda_i, \lambda_j); when is it positive semidefinite? Let A be a matrix whose entries are given by a polynomial,$$ A_{ij} = p(\lambda_i, \lambda_j) $$where p(\lambda_i,\lambda_j) = p(\lambda_j,\lambda_i) is symmetric. Are there standard methods ... 1 vote 1 answer 155 views ### Small linear relations between primitive Pythagorean triples \mathsf I Say a^2+b^2=c^2 is a primitive Pythagorean triple. Then consider the Linear Diophantine Equation$$ua^2+vb^2+xab+ybc+zca=0$$where (u,v,x, y, z)\in\mathbb Z^4 are variables. If (u,v,x, y, z)\neq(... 3 votes 1 answer 434 views ### Efficient computation of \sum_{i=1}^{\sqrt{n}} i^2\cdot\left\lfloor{\frac n{i^2}}\right\rfloor I need to compute efficiently the sum$$ \sum_{i=1}^{\sqrt{n}} i^2\cdot\left\lfloor{\frac n{i^2}}\right\rfloor. $$We can do this in O({\sqrt{n}}) but I need a faster algorithm: for example, it ... 0 votes 1 answer 190 views ### Write n^2 as x^2+y^2+2\times4^z or x^2+y^2+5\times 4^z In March 2018, I formulated the following somewhat curious question. Question 1. Whether for any integer n>1 there is a nonnegative integer k such that n^2-2\times 4^k or n^2-5\times 4^k ... 3 votes 1 answer 283 views ### How are natural numbers that cannot be written as a sum of exactly four squares of naturals characterized? [closed] This is most probably asked here already in some other form (as it seems to be a very basic question) but since I have't found some question very similar to this one I feel obliged to ask (also ... 9 votes 1 answer 290 views ### Hahn's approach to Hilbert's 17th problem? The Wikipedia article on Hahn Series mentions mentioned that these were studied by Hahn "in his approach to Hilbert's seventeenth problem". Is this correct? If so, what was this approach, ... 1 vote 0 answers 71 views ### Are those two Sum-Of-Squares approach for unconstrained polynomial optimization related? I found 2 approaches to solve an unconstrained polynomial optimization problem using the Lasserre / SOS hierarchy:$$ \inf_{x\in\mathbb{R}^n}\quad p(x), $$where p is a polynomial of even degree ... 2 votes 0 answers 175 views ### Sums of squares in fields Which fields k have the property that any sum of squares is a square ? Are there elegant characterizations and/or classifications known for this type of field ? And what if we replace "fields" by "... 1 vote 0 answers 93 views ### Lagrange's four squares theorem in other fields [duplicate] Is something known about analogues of Lagrange's four squares theorem in number fields other than \mathbb{Q}? I'm more interested in the case of finite extensions of \mathbb{Q}. For example, is ... 6 votes 1 answer 295 views ### Spherical Bessel functions. Sum of squares In (1) there is a property of spherical Bessel functions, which's derivation I can not find in the literature. {\mathsf{j}_{n}^{2}}\left(z\right)+{\mathsf{y}_{n}^{2}}\left(z\right)=\sum_{k=% 0}^{n}\... 4 votes 0 answers 480 views ### Four-square Conjecture Lagrange's four-square theorem states that every nonnegative integer can be written as the sum of four squares. My following conjecture is much stronger than this classical theorem. Four-square ... -1 votes 1 answer 356 views ### Positive integers written as \frac{a(a+1)}2+\frac{b(b+1)}2+4^c5^d Let \mathbb N=\{0,1,2,\ldots\}. Those T_n:=n(n+1)/2 with n\in\mathbb N are called triangular numbers. It is well known that$$\{T_a+T_b+T_c:\ a,b,c\in\mathbb N\}=\mathbb N\tag{1}$$which was ... 1 vote 1 answer 148 views ### Constrained optimization of sum of squares polynomials Consider the problem$$ \min p(x) \text{ subject to } g_j(x)\le 0 \quad p,g_j\in\text{SOS}, \qquad (*) $$i.e. p,g_j (j=1,\ldots,m) are sum of squares (SOS) polynomials. Can this problem be ... 2 votes 0 answers 39 views ### Sums of squares lengths in coordinates rings of plane curves let C\subseteq\mathbb{P}^2 be a plane algebraic curve and let \mathbb{R}[C] be its real coordinate ring. Let d\geq 1 and let p(d) be the smallest number such that every sum of squares in \... 18 votes 1 answer 712 views ### Lagrange four-squares theorem --- deterministic complexity Lagrange's four-squares theorem states that every natural number can be represented as the sum of four integer squares. Rabin and Shallit gave a randomised algorithm that finds one of these solutions ... 9 votes 1 answer 850 views ### Sums of two squares in arithmetic progressions Let r(n) denote the number of representations of n as the sum of two squares. Are there any known results on$$\sum _{n\leq x\atop {n\equiv a(q)}}r(n)$$and in particular is there an asymptotic ... 0 votes 0 answers 114 views ### Does positivstellensatz and SOS proof system help here? I have a system of m homogeneous degree 2 polynomial equations in \mathbb Z[x_1,\dots,x_n] where m=poly(n). Take$$f_1(x_1,\dots,x_n)=0\dotsf_m(x_1,\dots,x_n)=0 to be the system. ...
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 ...
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, ...