  The cancellation problem asks whether $A[X_1,X_2,\ldots,X_n] \cong
B[Y_1,Y_2,\ldots,Y_n]$ implies $A \cong B$. Hamann introduced the class of
steadfast rings as the rings for which a version of the cancellation problem
considered by Abhyankar, Eakin, and Heinzer holds. By work of Asanuma, Hamann,
and Swan, steadfastness can be characterized in terms of $p$-seminormality,
which is a variant of normality introduced by Swan. We prove that
$p$-seminormality and steadfastness deform for reduced Noetherian local rings.
We also prove that $p$-seminormality and steadfastness are stable under
adjoining formal power series variables for reduced (not necessarily
Noetherian) rings. Our methods also give new proofs of the facts that normality
and weak normality deform, which are of independent interest.
