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Uniform determinantal representations. (English) Zbl 1387.13062
Let $$K$$ be a field and $$d,n\in \mathbb Z_{\geq 0}$$. Let $$K[x_1,\ldots, x_n]$$ be the polynomial ring over $$K$$, and let $$p_{n,d} = \sum_{\alpha} c_{\alpha} x^{\alpha}$$ be a polynomial of degree at most $$d$$. Here $$\alpha = (\alpha_1,\ldots,\alpha_n) \in \mathbb Z^n_{\geq 0}$$ is a multi-index such that $$\sum_{i=1}^n \alpha_i\leq d$$, and $$x^{\alpha}$$ denotes the monomial $$\prod_{i=1}^n x_i^{\alpha_i}$$. A determinantal representation of $$p_{n,d}$$ is an $$N\times N$$-matrix $$M$$ with affine-linear entries in $$x_1,\ldots, x_n$$, such that $$\det(M) = p$$; the integer $$N$$ is the size of the representation. The minimal possible size is called determinantal complexity of $$p$$. Recently, this notion has become fundamental due to its connections to several fields such as complexity theory, optimization, and scientific computing. For instance, one of the deepest conjectures in algebraic complexity is Valiant’s conjecture, which states that the permanent of an $$m\times m$$-matrix does not admit a determinantal representation of size polynomial in $$m$$. (This conjecture can be also rephrased in the context of representation theory and orbit closures of permanents and determinants.) In this interesting paper, the authors study a variant of determinantal representations. Instead of looking at representations of a single polynomial, they seek for determinantal representations of subspaces of polynomials. A uniform determinantal representation of $$p_{n,d}$$ is an $$N\times N$$-matrix $$M$$ with entries in $$x_1,\ldots, x_n, c_{\alpha}$$, of degree at most one in each of these two sets of variables, such that $$\det(M) = p_{n,d}$$. Such a matrix $$M$$ gives a representation for all polynomials of degree at most $$d$$. For $$n,d\in \mathbb Z_{\geq 0}$$, the number $$N^{*}(n,d)\in \mathbb Z_{>0}$$ denotes the minimum size of uniform determinantal representations of $$p_{n,d}$$. Let $$M$$ be a uniform determinantal representation of $$p_{n,d}$$. Let $$M = M_0+M_1$$, where $$M_0$$ is the matrix not containing any variable $$c_{\alpha}$$. They show that for every point $$\overline{x}=(\overline{x_1},\ldots, \overline{x_n})\in K^n$$, $$M_0(\overline{x})$$ is singular (Lemma 2.5). Thus, they nicely connect the theory of uniform representations with the classical theory of singular matrix spaces; the connection is explained in Section 3. The main result is the following. For every integer $$n\geq 2$$, there exist positive constants $$C_1,C_2$$, depending on $$n$$, such that $$C_1 d^{n/2} \leq N^{*}(n,d)\leq C_2 d^{n/2}$$ (Theorem 1.3). These bounds improve previous results by R. Quarez [Linear Algebra Appl. 436, No. 9, 3462–3660 (2012; Zbl 1238.47010)]. They show explicit values of $$N^{*}(n,d)$$ for small $$n$$ and $$d$$. They prove that $$N^{*}(2,2) = 3$$ (Proposition 5.1) and $$N^{*}(3,2) = 4$$ (Proposition 5.2). Section 7 is the computational part of this article. Here they apply the results of the paper to the problem of efficiently solving systems of bivariate polynomial equations, based on the work of B. Plestenjak and M. E. Hochstenbach [SIAM J. Sci. Comput. 38, No. 2, A765–A788 (2016; Zbl 1376.65056)]. The authors conclude presenting several questions arising from this work. Finally, appendix A describes an algorithm to compute a determinantal representation for a given polynomial, based on the proof of Theorem 1.3.

##### MSC:
 13P15 Solving polynomial systems; resultants 65H04 Numerical computation of roots of polynomial equations 65F15 Numerical computation of eigenvalues and eigenvectors of matrices 65F50 Computational methods for sparse matrices
##### Software:
Bertini; BertiniLab; BiRoots; Mathematica; MultiParEig; NACLab; PHClab; PHCpack; rootsb
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