A basic fact in linear algebra is that the image of the curve
f(x)=(x^1,x^2,x^3,...,x^m), say over C, is not contained in any
m-1 dimensional affine subspace of C^m. In other words, the image
of f is not contained in the image of any polynomial-mapping
G:C^{m-1} ---> C^m of degree 1(that is, an affine mapping).
Can one give an explicit example for a polynomial curve
f:C ---> C^m, such that, the image of f is not contained in
the image of any polynomial-mapping G:C^{m-1} ---> C^m of degree 2 ?

We show that problems of this type are closely related to proving
lower bounds for the size of general arithmetic circuits.
For example, any explicit f as above of degree up to 2^{m^{o(1)}},
implies super-polynomial lower bounds for computing the permanent over C.

More generally, we say that a polynomial-mapping f:F^n ---> F^m is
(s,r)-elusive, if for every polynomial-mapping G:F^s ---> F^m of degree r, Image(f) is not contained in Image(G). We show that for many settings of the parameters n,m,s,r$, explicit
constructions of elusive polynomial-mappings imply strong (up
to exponential) lower bounds for general arithmetic circuits.

Finally, for every r < log n, we give an explicit example
for a polynomial-mapping f:F^n ---> F^{n^2}, of degree O(r),
that is (s,r)-elusive for s = n^{1+\Omega(1/r)}.
We use this to construct for any r, an explicit example
for an n-variate polynomial of total-degree O(r), such that,
any depth r arithmetic circuit for this polynomial (over any field)
is of size > n^{1+\Omega(1/r)}.

In particular, for any constant r, this gives a constant degree
polynomial, such that, any depth r arithmetic circuit for this
polynomial is of size > n^{1+\Omega(1)}.
Previously, only lower bounds of the type
\Omega(n \cdot \lambda_r (n)), where \lambda_r (n) are extremely
slowly growing functions (e.g., \lambda_5(n) = \log^{*}n, and
\lambda_7(n) = \log^* \log^{*}n), were known for constant-depth arithmetic circuits for polynomials of constant degree.