Let $\Delta$ be a standard $k$-simplex, and let $f:\Delta\to\mathbb R^N$ be a polynomial map with known numerical coefficients. What sort of practical computational algorithms can be used to investigate whether $f$ has self-intersections, or, the limiting case of self-intersection, points $x\in\Delta$ where the derivative $f'(x):\mathbb R^k\to\mathbb R^n$ has rank less than $k$? I'm not interested in the location of the points, and my wish is for a Yes/No answer.
However, one can't hope for a Yes/No answer. For example, if one is looking at real curves in $\mathbb R^3$, then changing one of the coefficients slightly will convert a near miss into an intersection, and floating point error will raise its ugly head. But one could hope for Yes/No/Near. It seems that one could do a brute force search, but this might take a long time. Can one do better than using one of the optimizing procedures for more general functions? I know little about this, but these general methods can miss the points one is looking for. Is there a method that takes advantage of the polynomial nature of $f$ to work faster and/or give better information? It seems fairly obvious that a brute force method could give a guaranteed Yes/No/Near answer, where Near means to within a given $\varepsilon$.
I'm mainly thinking of low degree polynomials, for example total degree 3, but I'm not interested in total degree 1, for which the answers are trivial. Even so, there can be a lot of coefficients, depending on the sizes of $k$ and $N$. I would like to implement something (or preferably use someone else's code) that works reasonably for $k\le10$, or even $k\le5$, and $N\le 50$. I'm also interested in a theoretical algorithm for the general case, without restrictions on $k$ and $N$, together with complexity estimates.