A real \(m\times n\) matrix is said to be nonnegative when its entries are nonnegative. We denote by rank\((A)\) the rank (i.e., the column space) of a matrix \(A\). Let \(A\) be a nonnegative matrix. The nonnegative rank of \(A\) is the smallest \(k\) for which \(A\) can be written as a product \(A=BC\) of two nonnegative matrices \(B\) and \(C\) of dimensions \( m\times k\) and \(k\times n \), respectively. Let \(\mathbb{F}\) be a subfield of \(\mathbb{R}\) and let \(\operatorname{rank}_{+}(A,\mathbb{F})\) denote the smallest integer \(k\) such that \(A\) can be written as a sum of \(k\) rank-one matrices with nonnegative entries in \(\mathbb{F}\). Note that \(\operatorname{rank}_{+}(A,\mathbb{R})\) corresponds to the nonnegative rank of \(A\).
The author provides an example of a subfield \(\mathbb{F\subset R}\) and a nonnegative matrix \(A\) with entries in \(\mathbb{F}\) such that \[ 5 = \operatorname{rank}(A) = \operatorname{rank}_{+}(A,\mathbb{R}) < \operatorname{rank}_{+}(A,\mathbb{F} ) = 6. \tag{\(\ast\)} \]
To show the existence of such a matrix \(A\), the author proves the following:
Theorem. There exists a field \(\mathbb{F\subset R}\) and polytopes \(P,Q\subset \mathbb{R}^{4}\) such that:

(1)

The vertices of both \(P\) and \(Q\) have coordinates in \(\mathbb{F}\);

(2)

\(\dim P=4\) and \(P\subset Q\);

(3)

There is a \(4\)-simplex \(\Delta \) satisfying \(P\subset \Delta \subset Q\);

(4)

Every such \(\Delta \) has a vertex where not all of whose coordinates belong to \(\mathbb{F}\).

The paper has five sections. In the introduction, the author gives a motivation and explains why the theorem above implies the existence of \(A\) as in (\(\ast\)). In Section 2, the author computes several numbers used in the description of the polytopes \(P\), \(Q,\) and \(\Delta \). In Section \(3\), the author describes these polytopes and proves items (1)–(3) of the above theorem. The author concludes the proof of the theorem in Section 4. Since the proof of the theorem requires some computer calculations, the author often refers in Sections 2–4 to given Wolfram Mathematica files attached to the current version of this paper on arXiv [the author, “Nonnegative rank depends on the field”, Preprint, arXiv:1505.01893]. The author concludes the paper with an open problem.