# Properties of functions with $0$ second partial derivatives

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!

• Perhaps $f$ is a polynomial, of degree $\;\le 1$ in each variable. For example, when $n=2$, $f(x,y) = a + bx + cy + dxy$. – GEdgar Apr 5 at 21:45
• 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 – wjmccann Apr 5 at 22:32

## 1 Answer

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 $$p(x_1^{\alpha_1},\dots,x_n^{\alpha_n})=c.$$