All Questions
Tagged with sums-of-squares ag.algebraic-geometry
7 questions with no upvoted or accepted answers
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
3
votes
0
answers
153
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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 ...
3
votes
0
answers
82
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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 ...
- 129
2
votes
0
answers
41
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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
2
votes
0
answers
246
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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 $c_i$...
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