  We obtain bounds on fractional parts of binary forms of the shape
$$\Psi(x,y)=\alpha_k x^k+\alpha_l
x^ly^{k-l}+\alpha_{l-1}x^{l-1}y^{k-l+1}+\cdots+\alpha_0 y^k$$ with
$\alpha_k,\alpha_l,\ldots,\alpha_0\in\mathbb{R}$ and $l\leq k-2.$ By exploiting
recent progress on Vinogradov's mean value theorem and earlier work on
exponential sums over smooth numbers, we derive estimates superior to those
obtained hitherto for the best exponent $\sigma$, depending on $k$ and $l,$
such that \begin{equation*} \min_{\substack{0\leq x,y\leq X\\(x,y)\neq
(0,0)}}\|\Psi(x,y)\|\leq X^{-\sigma+\epsilon}.\end{equation*}
