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Other gear you’ll need:

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https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2757311/

http://onlinelibrary.wiley.com/doi/10.1111/j.1476-5381.2011.01238.x/full

https://www.sciencedaily.com/releases/2007/08/070816094649.htm

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3048583/

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2503660/

cannot

A blog about probability and olympiads by Dominic Yeo

Posted on by dominicyeo

I’ve been taking a TCC course this term on Additive Combinatorics , delivered via video link from Bristol by Julia Wolf. At some point once the dust of this term has settled, I might write some things about the ideas of the course I’ve found most interesting, in particular the tools of discrete Fourier analysis to get a hold on some useful combinatorial properties of subsets of for example.

For this post, I want to talk instead about a topic that was merely mentioned in passing, the Combinatorial Nullstellensatz. The majority of this post is based on Alon’s original paper, which can be found , and Chapter 9 of Tao and Vu’s book Additive Combinatorics . My aim is to motivate the theorem, give a proof, introduce one useful application from additive combinatorics, and solve Q6 from IMO 2007 as a direct corollary.

What does Nullstellensatz mean? Roughly speaking, it seems to mean ‘a theorem specifying the zeros’. We will be specifying the zeros of a polynomial. We are comfortable with how the zeros of a complex-valued polynomial of one variable behave. The number of zeros is given precisely by the degree of the polynomial (allowing appropriately for multiplicity). It is generally less clear how we might treat the zeros of a polynomial of many variables. The zero set is likely to be some surface, perhaps of dimension one less than the number of variables. In particular, it no longer really makes sense to talk about whether this set is finite or not. The Combinatorial Nullstellensatz gives us some control over the structure of this set of zeros.

The idea behind the generalisation is to view the Fundamental Theorem of Algebra as a statement not about existence of roots, but rather about (combinatorial) existence of non-roots. That is, given a polynomial P(x) of degree n, for any choice of (n+1) complex numbers, at least one of them is not a root of P. This may look like a very weak statement in this context, where we only expect finitely many roots anyway, but in a multivariate setting it is much more intuitively powerful.

Recall that the degree of a monomial is given by the sum of the exponents of the variables present. So the degree of is 6. The degree of a polynomial is then given by the largest degree of a monomial in that polynomial. A polynomial over a field F with degree d might have lots of monomial terms of degree d. Suppose one of these monomials is , where . Then one version of the Combinatorial Nullstellensatz asserts that whenever you take subsets of the base field with , then there is a point with such that .

In other words, you can’t have a box (ie product of sets) of dimension on which the polynomial is zero.

Unsurprisingly, the proof proceeds by induction on the number of variables. Alon’s result proceeds via a more general theorem giving information about the possibility of writing multinomial polynomials as linear combinations of polynomials in one variable.

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