The maximal minors of a p by (m+p)-matrix of univariate
polynomials of degree n with indeterminate coefficients
are themselves polynomials of degree np.
The subalgebra generated by their coefficients is
the coordinate ring of the quantum Grassmannian,
a singular compactification of the space of rational curves
of degree np in the Grassmannian of p-planes in
(m+p)-space.
These subalgebra generators are shown to form a sagbi basis.
The resulting flat deformation from the quantum Grassmannian
to a toric variety gives a new "Gröbner basis style" proof of
the Ravi-Rosenthal-Wang formulas in quantum Schubert calculus.
The coordinate ring of the quantum Grassmannian is an
algebra with straightening law, which is normal,
Cohen-Macaulay, Gorenstein and Koszul,
and the ideal of quantum Plücker relations has a
quadratic Gröbner basis. This holds more
generally for skew quantum Schubert varieties. These results
are well-known for the classical Schubert varieties (n=0).
We also show that the row-consecutive p by p-minors of
a generic matrix form a sagbi basis
and we give a quadratic Gröbner basis for their algebraic relations.