Algebra/Linear Algebra Seminar

Speaker: Luke Oeding

Title: Border ranks of monomials

Abstract: What is the minimal number of terms needed to write a monomial as a sum of powers? What if you allow limits? Here are some minimal examples:

\(4xy = (x+y)^2 - (x-y)^2\)

\(24xyz = (x+y+z)^3 + (x-y-z)^3 + (-x-y+z)^3 + (-x+y-z)^3 192xyzw = (x+y+z+w)^4 - (-x+y+z+w)^4 - (x-y+z+w)^4\)

\(- (x+y-z+w)^4 - (x+y+z-w)^4  + (-x-y+z+w)^4 + (-x+y-z+w)^4 + (-x+y+z-w)^4\).

The monomial x^2y has a minimal expression as a sum of 3 cubes:

\(6x^2y = (x+y)^3 + (-x+y)^3 -2y^3\).

But you can use only 2 cubes if you allow a limit:

\(6x^2y = lim_{\epsilon \to  0} \frac{(x^3 - (x-\epsilon y)^3)}{\epsilon}\).

Can you do something similar with xyzw? Previously it wasn't known whether the minimal number of powers in a limiting expression for xyzw was 7 or 8.

I will answer this and the analogous question for all monomials.

The polynomial Waring problem is to write a polynomial as linear combination of powers of linear forms in the minimal possible way. The minimal number of summands is called the rank of the polynomial. The solution in the case of monomials was given in 2012 by Carlini, Catalisano, and Geramita, and independently shortly thereafter by Buczynsk, Buczynski, and Teitler. In this talk I will address the problem of finding the border rank of each monomial.

Upper bounds on border rank were known since Landsberg-Teitler, 2010, and earlier. We use symmetry-enhanced linear algebra to provide polynomial certificates of lower bounds (which agree with the upper bounds).

This work builds on the idea of Young flattenings, which were introduced by Landsberg and Ottaviani, and give determinantal equations for secant varieties and provide lower bounds for border ranks of tensors.  We find special monomial-optimal Young flattenings that provide the best possible lower bound for all monomials up to degree 6. For degree 7 and higher these flattenings no longer suffice for all monomials. To overcome this problem, we introduce partial Young flattenings and use them to give a lower bound on the border rank of monomials which agrees with Landsberg and Teitler's upper bound. I will also show how to implement Young flattenings and partial Young flattenings in Macaulay2 using Steven Sam's PieriMaps package.