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is multilinear, agrees with at every point of , is degree-nonincreasing when nonzero, and is the unique multilinear polynomial with that agreement
Statement
For every polynomial , its multilinear reduction is multilinear, agrees with at every point of the cube , satisfies whenever , and is the unique multilinear polynomial with that agreement.
Facts & Assumptions
Given: a polynomial .
The multilinear reduction replaces each monomial by the squarefree monomial (Multilinear polynomials and the reduction on the cube).
A polynomial with each variable degree below that vanishes on the whole cube is the zero polynomial (If for each and vanishes on , then ).
Proof
For and every positive integer , . Hence each monomial and its reduction have the same value at , so summing the monomials in [F1] gives . Every reduced monomial is squarefree and has degree , so is multilinear and has no larger total degree than whenever the two are nonzero.
If is another multilinear polynomial agreeing with on the cube, then is multilinear and vanishes on the cube. By [L2], it is the zero polynomial. So , proving uniqueness.
Remarks
- Uniqueness is what later turns a pointwise identity on the cube into a linear independence statement about multilinear monomials.
Depends on
- Multilinear polynomials and the reduction $x_i^{2}\mapsto x_i$ on the cube
- Reducing $f$ modulo $g_i(x_i)=\prod_{s\in S_i}(x_i-s)$ lowers each $\deg_{x_i}$ below $\lvert S_i\rvert$, preserves the values on the grid, and preserves any top-degree coefficient whose exponents stay below the grid sizes
- If $\deg_{x_i}P<\lvert S_i\rvert$ for each $i$ and $P$ vanishes on $S_1\times\cdots\times S_n$, then $P=0$
- Monomials, coefficients, degree in each variable and total degree in $F[x_1,\dots,x_n]$
- Evaluation and roots of a polynomial in a commutative target ring
Used by
- The six 2-subsets of [4] are {0,1}-intersecting, and the bound ∑_i≤2C(4, i)=11 holds Example
- The functions {0,1}ⁿ→ F obtained from x_T with | T|≤ s are linearly independent, so they span a space of dimension ∑ᵢ₌₀ˢC(n, i) Lemma
- A second proof of Sauer–Shelah, from the multilinear polynomial space Theorem
- An L-intersecting family on [n] with | L|=s has at most ∑ᵢ₌₀ˢC(n, i) members Theorem
Dependency tree · two levels
16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- J. Matousek, Thirty-three Miniatures, Miniature 17 (standard reference, not scraped)
- L. Babai and P. Frankl, Linear Algebra Methods in Combinatorics, §4.3 (standard reference, not scraped)