A visual essay
Polynomial identities, visually
What does it mean for a polynomial to vanish on every matrix? Eight pictures, from a single commutator to a database of tens of thousands of certified values.
Waldeck Schützer · UFSCar
Your system asks for reduced motion: the pictures change with your scrolling but do not animate by themselves.
Scene 1 of 8
Order matters
For numbers, \(ab-ba=0\) always. For matrices it almost never is: multiply the same two matrices in the other order and the picture changes. A polynomial that still vanishes whatever we plug in is called an identity, and the commutator is not one.
Scene 2 of 8
A polynomial every 2×2 matrix obeys
Take four matrices and add up all 24 ways to multiply them in a row, each with a plus or minus sign depending on the order. This is the standard polynomial \(s_4\). Amitsur and Levitzki proved in 1950 that for \(2\times2\) matrices it is always the zero matrix. Watch the running total return to zero.
Scene 3 of 8
Every degree has its shapes
Identities of degree \(d\) live inside the \(d!\) ways to order \(d\) variables. Shuffling the variables splits that space into blocks, one for each way to stack \(d\) boxes into a Young diagram. For \(d=4\) there are five shapes, and their blocks have sizes \(1,3,2,3,1\): the squares add up to \(24\).
Scene 4 of 8
A worked example: 2×2 matrices, degree 4
How many identities hide in each block? In the database, for \(2\times2\) matrices in degree 4, four blocks contain none and the column-shaped one contains exactly one copy: it is spanned by \(s_4\). Every degree-4 identity of \(M_2\) is a multiple of the standard polynomial.
Scene 5 of 8
How a computer finds them
Plug matrix units (matrices with a single 1) into the 24 orderings. Each choice gives a column of numbers; adding up their products with themselves builds a \(24\times24\) table. Its null directions are the identities. The rank counts what is not an identity. Watch the rank climb as more choices are added.
Scene 6 of 8
A map of what is known
Each square is one matrix size (rows) and one degree (columns). Green means every block of that cell has been pinned down exactly, with a certificate. Amber means some blocks are only bracketed between a lower and an upper bound. The database has tens of thousands of such values.
Scene 7 of 8
How sure are we?
Every value carries a status. A lower bound comes from an identity we can exhibit; an upper bound from a nonzero determinant. When they meet, the value is exact. If a randomized test is the only evidence the value is marked probable, and it is never promoted to exact.
Scene 8 of 8
What remains
The frontier is the amber part of the map: the cells where some blocks are still only bracketed between two bounds that have not yet met. Closing them needs either a new family of identities or a faster certificate. Everything on this page can be explored, cell by cell, in the database.