The Segre map
On first hand, the Segre map is the function
$$\sigma_{n,m} : \mathbb{P}^n \times \mathbb{P}^m \rightarrow \mathbb{P}^{(n+1)(m+1)-1} : ([x_0,...,x_n],[y_0,...,y_m]) \rightarrow [x_0y_0, x_0y_1,...,x_0y_m,x_1y_0,...,x_iy_j,...,x_ny_m]$$
and the Segre variety $\Sigma_{n,m}$ is the image of the Segre map $$\Sigma_{n,m}=\sigma_{n,m}(\mathbb{P}^n \times \mathbb{P}^m).$$
Let $Z_{i,j}=x_iy_j$. The Segre variety is also defined as the common zero locus of the quadratic polynomials $$Z_{{i,j}}Z_{{k,l}}-Z_{{i,l}}Z_{{k,j}},$$ where $Z_{i,j}$ is understood to be the natural coordinate on the image of the Segre map.
The k-secant of a projective variety
The k-secant of $X\subset \mathbb{P}^n$ is $$ \sigma_k(X)=\overline{\underset{x1,...,x_k\in X}{\cup} \mathbb{P}^{k-1}_{x_1,...,x_k}} $$ where $\mathbb{P}^{k-1}_{x_1,...,x_s}$ is a projective space of dimension $s-1$ passing through $x_1,...,x_s$. If needed to better understand this definition, another equivalent approach to define k-secant is the following.
Let $Y\subset \mathbb{P}^n$. The joint of X and Y is $$J(X,Y)=\overline{\underset{x\in X, y\in Y, x\neq y}{\cup} \mathbb{P}^1_{x,y}}$$ where $\mathbb{P}^1_{x,y}$ is the projective line containing x and y.
The 2-secant of X, also simply named the secant of X, is $ \sigma(X)=J(X,X) $ and the k-secant is $$\sigma_k(X)=J(X,\sigma_{k-1}(X)).$$
Question
What is the set of polynomials whose common zero locus is the k-secant of the Segre variety ?
