Explicit tensors or polynomials of general rank.

The Waring rank of a general polynomial (meaning, general coefficients), in a given number of variables and of a given degree, has been known since 1995 (work of Alexander-Hirschowitz). A random polynomial has this general rank with probability $1$ (with respect to any reasonable meaning of “random polynomial”). But there’s no known way to get an explicit polynomial of general rank beyond some small cases. Nor is there any good way to check if a given polynomial indeed has general rank.

Fixing a degree $d$ and letting the number of variables grow, the general rank is a polynomial of degree $d-1$ (after finitely many exceptional cases). There’s no known explicit sequence of polynomials $p_n$, of degree $d$ in $n$ variables, with rank even growing super-linearly, let alone of order $n^{d-1}$ (ignore $d=2$).

I believe all of this holds as well for tensors, with the exception that in most tensor products the general rank isn’t known yet.

Explicit tensors or polynomials of general rank.

The Waring rank of a general polynomial (meaning, general coefficients), in a given number of variables and of a given degree, has been known since 1995 (work of Alexander-Hirschowitz). A random polynomial has this general rank with probability $1$ (with respect to any reasonable meaning of “random polynomial”). But there’s no known way to get an explicit polynomial of general rank beyond some small cases. Nor is there any good way to check if a given polynomial indeed has general rank.

Fixing a degree $d$ and letting the number of variables grow, the general rank is a polynomial of degree $d-1$ (after finitely many exceptional cases). There’s no known explicit sequence of polynomials $p_n$, of degree $d$ in $n$ variables, with rank growing with order $n^{d-1}$ (ignore $d=2$). (Elementary symmetric polynomials of odd degree achieve order $n^{(d-1)/2}$.)

I believe all of this holds as well for tensors, with the exception that in most tensor products the general rank isn’t known yet.