Brian Wecht
speaker
852 appearances
2 recordings
1 series
first heard Aug 2026
last heard 5 Sep
Brian Wecht’s voice in public audio — every appearance, attributed to the second.
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recordings per month · last 12 monthsRecordings per month over the last 12 months — 2 in all, peaking in Sep 2026 with 1.
Appearances
A polynomial in one variable, let's say x, is just when you get when you take x, you raise it to different powers, different integer powers.
x, x squared, x cubed, x to the fourth, etc., and you add or subtract those together with different coefficients in front of them.
Those are polynomials in that's a polynomial in one variable.
Polynomials in two variables or three variables or four variables, et cetera, are exactly the same kind of thing.
All you do or you multiply the different number of variables together, you can raise them to integer powers, and you can add them all together.
It's basically the simplest kind of function
Function that you can think of.
And you can write down polynomials in any number of variables you want.
So the Jacobian conjecture has to do with functions from three variables to three variables, four variables to four variables, et cetera, et cetera.
Right?
Same number of variables to itself.
And if you like, it's like mapping a plane, which would be two variables to another plane, a three-dimensional space to another three-dimensional space, and so
Mm-hmm.
So the Jacobian conjecture says that anytime I can write down a polynomial function in n variables, if it satisfies a certain uh property, and I'm just gonna say the words and then I'll talk very briefly about what they mean, which is that the Jacobian determinant is a constant.
And basically this is some kind of smoothness of the of the map, some kind of smoothness of the function.
that I can take a map from these three variables, or n variables, I should say, to these n variables over here.
satisfies a certain condition that I can always find a polynomial that reverses it that goes from where I ended up back to the original.
Basically I can invert my original thing.
So if I know where I started, I know where I end up, and if I know where I ended up, I know where I started.
And that's not always true.
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