All Questions
Tagged with sums-of-squares ag.algebraic-geometry
16 questions
3
votes
0
answers
153
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 ...
11
votes
0
answers
363
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
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. ...
- 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 ...
- 1,826
3
votes
0
answers
82
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 ...
- 129
2
votes
0
answers
41
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
0
votes
0
answers
127
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.
...
- 1,826
0
votes
0
answers
129
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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 ...
- 1,826
4
votes
1
answer
113
views
Specific quaternary quartic that is positive semi-definite but not sum of squares
Does there exist a quaternary quartic $f$ (a form in $\mathbb{R}[x_1,x_2,x_3,x_4]$ of degree $4$), which is positive semi-definite ($f \geq 0$ on $\mathbb{R}^4$) but not a sum of squares, such that ...
- 41
5
votes
1
answer
265
views
polynomial maps from reducible plane curves to conics
It is classically known that every smooth plane quartic curve $C$ can be represented by an equation $q_1 q_3 = q_2^2,$ with $q_j\in\mathbb{C}[X,Y,Z]$, $1\leq j\leq 3$ quadratic forms, and the same is ...
- 14k
6
votes
1
answer
229
views
For a Sum-of-Squares cost functions J(x) is it true that J(x)-j* is also SOS?
For polynomial optimization problems the sum-of-squares theory and Lasserre relaxation hierarchy provides a theoretically handy way of getting the solution. There are also results saying that finite ...
- 145
2
votes
0
answers
246
views
Sum of Squares and Harmonic Functions
Let $a_0, a_1, \dots a_k$ be non-negative reals. Any homogeneous polynomial $p$ of degree $2k$ in $\mathbb{R}^{d}$ can be decomposed as
$$
p(x)=\sum_{i=0}^{k}c_i|x|^{2(k-i)}p_{2i}(x)
$$
for some $c_i$...
4
votes
1
answer
830
views
quartic diagonal as a sum of squares of quadratic forms
I would appreciate if someone can point out to the literature related to characterizing the set of all different ways to write real quartic diagonal $\sum \limits_{k=1}^n x_k^4, x \in \mathbb{R^n}$ as ...
- 63
7
votes
1
answer
389
views
Is this Negativstellensatz with uniform denominators known?
A theorem of Reznick states that if $f>0$ is a real homogeneous polynomial in several polynomials that is positive away from the origin of ${\mathbb{R}}^n$, then for large $N$, the form $(\sum x_i^...
user2529
3
votes
3
answers
958
views
solutions to equation mod a prime
I know that characterizing the solutions to an equation in a finite field is generally difficult, but I was wondering if anyone had anything to say about the equation
(ab)^2 + a^2 + b^2 = 0 mod p
I ...
- 39
15
votes
3
answers
2k
views
Polynomials that are sums of squares
Is any algorithm known for determining whether or not a multivariate polynomial with integer coefficients can be written as a sum of squares of such polynomials?
By way of background, if we one ...
- 151