  Assuming that the Permanent polynomial requires algebraic circuits of
exponential size, we show that the class VNP does not have efficiently
computable equations. In other words, any nonzero polynomial that vanishes on
the coefficient vectors of all polynomials in the class VNP requires algebraic
circuits of super-polynomial size.
  In a recent work of Chatterjee and the authors (FOCS 2020), it was shown that
the subclasses of VP and VNP consisting of polynomials with bounded integer
coefficients do have equations with small algebraic circuits. Their work left
open the possibility that these results could perhaps be extended to all of VP
or VNP. The results in this paper show that assuming the hardness of Permanent,
at least for VNP, allowing polynomials with large coefficients does indeed
incur a significant blow up in the circuit complexity of equations.
