### Positive polynomial as a sum of squares

I found a question to prove the polynomial $58x^{10}-42x^9+11x^8+42x^7+53x^6-160x^5+118x^4+22x^3-56x^2-20x+74$ to be positive on $\mathbb{R}$. The solution was $(7x^5-4x^3+6x^2+2x-5)^2+(-3x^5+7x^4-3x^3+7)^2$. Is there a method in Sage to find such a representations? I tried the code modified from https://ask.sagemath.org/question/10062/polynomials-as-a-sum-of-squares/

```
R = AA[x]
def sum_of_two_squares(P):
r"""
P is assumed to be a polynomial defined on the Algebraic Real Field.
Returns False is P is not positive.
Returns a pair of polynomial (A,B) such that A^2 + B^2 = P otherwise
"""
# try to convert P if it is defined on a subfield of AA, say QQ.
if P.parent() != R:
P = P.change_ring(AA)
LC = P.leading_coefficient()
if LC < 0:
return False
# Q will be the part of P with real roots.
Q = R(1)
for i in P.roots():
if i[1] % 2 != 0:
return False
else:
Q = Q * (R(x)-i[0])^i[1]
if P == LC * Q:
return (sqrt(LC) * sqrt(Q),R(0))
T = R(1)
for fact,mult in R(P/Q).factor():
f = fact.change_ring(QQbar)
T = T * (R(x)-f.roots()[0][0])^mult
# extract real and imaginary part of T
RE = R(0)
IM = R(0)
for i in range(T.degree()+1):
RE += T[i].real()*R(x)^i
IM += T[i].imag()*R(x)^i
SLC = sqrt(LC)
SQ = sqrt(Q)
return (SLC*SQ*RE, SLC*SQ*IM)
R = AA[x]
P = R(58*x^10-42*x^9+11*x^8+42*x^7+53*x^6-160*x^5+118*x^4+22*x^3-56*x^2-20*x+74)
A,B = sum_of_two_squares(P)
A
7.615773105863908?*x^5 - 2.75743509005418?*x^4 - 44.2356981337271?*x^3 + 11.9854478203666?*x^2 + 25.5052589522359?*x - 3.05770635146248?
```

It looks like the code does not always find the simplest form of solution. So is there an algorithm that ~~return the ~~tries to find a representation ~~with ~~such that it first tries to represent the polynomial as a sum of two constants, then sum of two polynomials of degree at most 1, then sum of two polynomials of degree at most 2 and so on and in every case the one can see the minimal polynomial of the coefficients ~~are some "simple" form like integers or roots of some integer coefficient polynomial with degree not too large?~~over $\mathbb Q[x]$.