Intereting Posts

Integral $ \int_0^1 \frac{\ln \ln (1/x)}{1+x^{2p}} dx$…Definite Integral
Product of two power series
Exponential Diophantine equation $7^y + 2 = 3^x$
Equivalent conditions for a linear connection $\nabla$ to be compatible with Riemannian metric $g$
How to explain the formula for the sum of a geometric series without calculus?
Intuitive reasoning behind $\pi$'s appearance in bouncing balls.
What are the important properties that categories are really abstracting?
How to solve ODEs by converting it to Clairaut's form through suitable substitutions.
Solution of eikonal equation is locally the distance from a hypersurface, up to a constant
Is any closed ball non-compact in an infinite dimensional space?
Probability of having $k$ similar elements in two subsets.
Are Hilbert primes also Hilbert irreducible ? Furthermore, are Hilbert primes also primes in $\mathbb{ Z}$?
Evaluate Integral $\int_{c\ -\ j\infty}^{c\ +\ j\infty} \left({\sigma\,x^{-1}}\right)^s{\Gamma(\beta_1-1+s)\over \Gamma(\beta_1+\beta_2-1+s)}\,ds$
Calculating closest and furthest possible diagonal intersections.
Monotonicity of the sequences $\left(1+\frac1n\right)^n$, $\left(1-\frac1n\right)^n$ and $\left(1+\frac1n\right)^{n+1}$

Let $f(x)$ be a polynomial function with real coefficients such that $f(x)\geq 0 \;\forall x\in\Bbb R$. Prove that there exist polynomials $A(x),B(x)$ with real coeficients such that $f(x)=A^2(x)+B^2(x)\;\forall x\in\Bbb R$

I don’t know how to approach this, apart from some cases of specific polynomials that turned out really ugly. Any hints to point me to the right direction?

- weak convergence in $L^p$ plus convergence of norm implies strong convergence
- Monotonic, surjective function
- How to prove $\sum\limits_{k=1}^{\infty}|\alpha_{k}|\lt \infty$, given that $\sum \limits_{k=1}^{\infty}\alpha_{k}\phi_{k}$ converges …?
- Faulhaber's Formula to evaluate series
- Does there exist a bijective $f:\mathbb{N} \to \mathbb{N}$ such that $\sum f(n)/n^2$ converges?
- How to find the limit of the sequence $x_n =\frac{1}{2}$, if $x_0=0$ and $x_1=1$?

- $\frac{(2n)!}{4^n n!^2} = \frac{(2n-1)!!}{(2n)!!}=\prod_{k=1}^{n}\bigl(1-\frac{1}{2k}\bigr)$
- Increasing, continuous function implies connectivity and viceversa.
- limit of a recursively defined function
- If $\lim a_n = L$, then $\lim s_n = L$
- Showing an indentity with a cyclic sum
- Infinite series $\sum_{n=0}^{\infty}\arctan(\frac{1}{F_{2n+1}})$
- $f,g,h$ are polynomials. Show that…
- Does the frontier of an open set have measure zero (in $\mathbb{R}^n$)?
- Determine the coefficients of an unknown black-box polynomial
- Find the degree of the splitting field of $x^4 + 1$ over $\mathbb{Q}$

Consider roots of $f(x)$, as $f(x)\geq0,\forall x\in\mathbb{R}$, so $f(x)$ can be rewritten as following:

$$f(x)=a^2(x-a_1)^2\cdots(x-a_k)^2[(x-\alpha_1)(x-\bar{\alpha_1})]\cdots[(x-\alpha_l)(x-\bar{\alpha_l})]$$

Where $a,a_1,\cdots,a_k\in\mathbb{R},\alpha_1,\cdots,\alpha_l\in\mathbb{C}$.

Denote $g(x)=a(x-a_1)\cdots(x-a_k),h(x)=(x-\alpha_1)\cdots(x-\alpha_l)=h_1(x)+ih_2(x)$, then

\begin{align*}

f(x)&=g^2(x) \, h(x) \, \bar{h}(x)\\

&=g^2(x) \, [h_1(x)+ih_2(x)] \, [h_1(x)-ih_2(x)]\\

&=(g(x)h_1(x))^2+(g(x)h_2(x))^2

\end{align*}

Assume that the leading coefficient of $f$ is $1$. If $f(x)=x^k$ for some $k$, then $k$ is even and the result follows. By completing the square we see that the result holds if the degree of $f$ is less than or equal to $2$. If $f$ has a real root, we may assume this root is at 0 by replacing $x$ with $x-a$. We can see that this must be a multiple root by the fact that $f(x)\geq 0$ for all $x$, so by factoring out $x^2$ we reduce the problem to a polynomial of lower degree. We may thus assume that $\deg(f)>2$, the result holds for polynomials smaller degree, and that $f$ has a nonzero complex root. The nonzero roots of $f$ are $z_1,z_1′,z_2,z_2′,\ldots,z_n,z_n’$ where $z_i$ is the complex conjugate of $z_i’$ for all $i$. Thus there is a quadratic polynomial $x^2+bx+c=(x-z_1)(x-z_1′)$ and a polynomial $p(x)$ of degree two less than the degree of $f$ such that

$$f(x)=(x^2+bx+c)p(x)$$

By induction, $p(x)=A(x)^2+B(x)^2$ for some polynomials $A(x)$ and $B(x)$. Thus

$$f(x)=(x^2+bx+c)(A(x)^2+B(x)^2)$$

We also have that

$$f(x)=((x+\frac{b}{2})^2+c-\frac{b^2}{4})(A(x)^2+B(x)^2)$$

so

$$f(x)=\left((x+\frac{b}{2})A(x)+\sqrt{c-\frac{b^2}{4}}B(x)\right)^2+\left((x+\frac{b}{2})B(x)-\sqrt{c-\frac{b^2}{4}}A(x)\right)^2$$

by the Brahmaguptaâ€“Fibonacci identity, so the result follows by induction provided $c\geq b^2/4$. However, if this were not the case then $x^2+bx+c$ would have a real root, so we are done.

Survey article by Bruce Reznick called *Some Concrete Aspects of Hilbert’s 17th Problem*, includes your case in the paragraph on Before 1900:

- For groups $A,B,C$, if $A\times B$ and $A\times C$ are isomorphic do we have $B$ isomorphic to $C$?
- How to prove Fibonacci sequence with matrices?
- What does a condition being sufficient as well as necessary indicates?
- Cardinality of cartesian product
- Comaximal ideals
- About evaluating $\mathcal{L}^{-1}_{s\to x}\bigl\{\frac{F(s)}{s}\bigr\}$ by considering contour integration with different entire functions $F(s)$
- About idempotent and invertible matrix
- Solving for streamlines from numerical velocity field
- Prove with induction that $11$ divides $10^{2n}-1$ for all natural numbers.
- Characteristic of a field is $0$ or prime
- Bessel and cosine function identity formula
- Does A5 have a subgroup of order 6?
- If A, B, C, D are non-invertible $n \times n$ matrices, is it true that their $2n \times 2n$ block matrix is non-invertible?
- Trapezoid rule error analysis
- What are the quadratic extensions of $\mathbb{Q}_2$?