Let us assume that we have $n$ polynomials in $n$ variables, $p_1(\vec{x}), p_2(\vec{x}),\ldots p_n(\vec{x})$. Jacobian, $J(p)$ of $\vec{p}$ is a matrix with $i,j$ entry $\frac{\partial p_i}{\partial x_j}$ Assume that $p_i$ are algebraically dependent, i.e., there exists polynomial $f$ such that $f(p_1,\ldots, p_n)=0$ then using chain rule of derivatives one can get that $J(p)\cdot (\partial f(p))=0$.

Over the field of zero characteristic the converse is also true. If Jacobian is not of the full rank then polynomials are algebraicly dependent. My question is if the stronger claim is true:

Assume that $\vec{v}=(v_1(x),\ldots, v_n(x))$ is a vector of polynomials such that $J(p)v=0$, then there exists polynomial $f$, such that $(\partial f)(\vec{p})=\vec{v}$.