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Potentially this can be solved with QEPCAD package (which need to be installed first). Here is an example:

x,y = var('x y')
P = x^3 + y^2 + 1

qf = qepcad_formula.and_( P == 0, x >= 0, y >= 0 )
if qepcad( qf ) == 'FALSE':
   print('P has no root in R+')
else:
   print('P has a root in R+')

Potentially this can be solved with QEPCAD package (which need needs to be installed first). Here is an example:

x,y = var('x y')
P = x^3 + y^2 + 1

qf = qepcad_formula.and_( P == 0, x >= 0, y >= 0 )
if qepcad( qf ) qepcad(qf) == 'FALSE':
   print('P has no root in R+')
else:
   print('P has a root in R+')

Potentially this can be solved with QEPCAD package (which needs to be installed first). Here is an example:

x,y = var('x y')
P = x^3 + y^2 + 1

qf = qepcad_formula.and_( P == 0, x >= 0, y >= 0 )
if qepcad(qf) == 'FALSE':
    print('P has no root in R+')
else:
    print('P has a root in R+')

Potentially this can be solved with QEPCAD package (which needs to be installed first). Here is an example:

x,y = var('x y')
P = x^3 + y^2 + - 1

qf = qepcad_formula.and_( P == 0, x >= 0, y >= 0 )
if qepcad(qf) == 'FALSE':
try:
    root = qepcad(qf,solution='any-point')
except ValueError:
    print('P has no root in R+')
else:
    print('P print(f'P has a root in R+')
root: {root}')

It can be run at Sagecell.