I have a $n$-dimensional polynomial that I am evaluating on some domain $\Omega \subset \mathbb{R^n}$ $$ f:\Omega\rightarrow \mathbb{R} $$

where I know that all the second partials are zero $$ \dfrac{\partial^2 f}{\partial x_k^2} \equiv0 $$

however, the mixed partials may be non-zero.

It is easy to see that this function is harmonic as $\nabla^2f = 0$. From this we get lots of results, such as the minima $m$ and maxima $M$ is on the boundary: $m,M\in \partial\Omega$.

This condition that all second derivatives seems stronger than being harmonic though, so I was wondering if there was a name / other properties relating to functions like this.

Thank you!

  • $\begingroup$ Perhaps $f$ is a polynomial, of degree $\;\le 1$ in each variable. For example, when $n=2$, $f(x,y) = a + bx + cy + dxy$. $\endgroup$ – GEdgar Apr 5 at 21:45
  • $\begingroup$ That is the form of solutions I found for higher dimensions too, but I was wondering if that type of polynomial had a special name $\endgroup$ – wjmccann Apr 5 at 22:32

These polynomials are called multilinear. They are sometimes called polynomials of max degree $1$ (as opposed to total degree $1,$ which would mean affine.)

They come up naturally in Fourier analysis on $\{-1,1\}^n$: the Fourier transform passes between the representation of a multilinear polynomial by its values on $\{-1,1\}^n$ and the values of its coefficients.

They also come up in things like the polynomial method, Schwartz-Zippel lemma, and related results. The important property here is that if a multilinear polynomial $p$ is zero on a grid of the form $\{a_1,b_1\}\times\dots\times\{a_n,b_n\}$ with $a_i\neq b_i$ for each $i,$ then $p$ is identically zero.

Since you have an algebraic property of polynomials, the restriction to $\Omega$ is irrelevant. But it is true that any distribution on a domain $\Omega$ satisfying $\partial^2 f/\partial x_k^2=0$ must be a multilinear polynomial.

Another property, perhaps only interesting to me: they are "nonzero at one of their multi-indices". By this I mean that if $p$ is a non-zero multilinear polynomial, there exist $\alpha_1,\dots,\alpha_n\in\{0,1\}$ such that $p(\alpha_1,\dots,\alpha_n)\neq 0$ and the $x_1^{\alpha_1}\dots x_n^{\alpha_n}$ coefficient of $p$ is non-zero. Specifically, we can take $(\alpha_1,\dots,\alpha_n)$ to be minimal in the product order on $\{0,1\}^n,$ such that the coefficient $c$ of $x_1^{\alpha_1}\dots x_n^{\alpha_n}$ is non-zero. This gives $p(x_1^{\alpha_1},\dots,x_n^{\alpha_n})=c.$


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