The book Solving Systems of Polynomial Equations by Bernd Sturmfels claims that you can translate a system of polynomials into a system of partial differential equations. The example given is the equation:
$\sum ci_{1}i_{2}...i_{n}x_{1}^{il}x_{2}^{i2}...x_{n}^{in}$=0
can be translated to
$\sum ci_{1}i_{2}...i_{n} \frac{\partial^{i_{1} + i_{2} + ... + i_{n}}f}{\partial_{x1}^{i_{l}}\partial_{x2}^{i_{2}}...\partial_{xn}^{i_{n}}}$ =0
for an unknown function $f = f(x_1, . . . , x_n)$.
My question is where does this translation come from?