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Computing Mixed Volume and All Mixed Cells in Quermassintegral Time

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Abstract

The mixed volume counts the roots of generic sparse polynomial systems. Mixed cells are used to provide starting systems for homotopy algorithms that can find all those roots and track no unnecessary path. Up to now, algorithms for that task were of enumerative type, with no general non-exponential complexity bound. A geometric algorithm is introduced in this paper. Its complexity is bounded in the average and probability-one settings in terms of some geometric invariants: quermassintegrals associated with the tuple of convex hulls of the support of each polynomial. Besides the complexity bounds, numerical results are reported. Those are consistent with an output-sensitive running time for each benchmark family where data are available. For some of those families, an asymptotic running time gain over the best code available at this time was noticed.

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Abbreviations

\(V(\mathcal A_1, \dots , \mathcal A_n)\) :

Mixed volume of \(\mathcal A_1, \dots , \mathcal A_n\).

\(A_i\) :

Finite subset of \(\mathbb Z^n\).

V :

Scaled mixed volume \(n!V( \mathrm {Conv}(A_1), \ldots , \mathrm {Conv}(A_n))\).

\(V_i\) :

Generic root bound of an unmixed system of support \(A_i\).

\(T, T'\) :

Time bounds for the algorithm

s :

Number of different supports \(A_i\).

\(m_i\) :

Multiplicity of each support \(A_i\).

\(b_i = b_i(\mathbf a) = b(i, \mathbf a)\) :

Lifting value for \(\mathbf a \in A_i\).

\(\lambda _i = \lambda _i(\mathbf \xi )\) :

Legendre dual for the lifting \(b_i\).

\(m_i(\varvec{\xi })\) :

Number of times \(\lambda _i(\mathbf \xi )\) attained, minus one.

\(L_{i,\varvec{\xi }}\) :

Facet of \(\mathrm {Graph}(\hat{b}_i)\).

\(L_{\varvec{\xi }}\) :

Facet of \(\mathrm {Graph}(\sum t_i \hat{b}_i), t_i\) indeterminates.

\(\Xi (L)\) :

Possibly unbounded polyhedron dual to face L.

\(F_0 \subset \cdots \subset F_n\) :

Generic affine flag in \(\mathbb R^n\).

\({m_{i}^{(d)}}\) :

Certain non-decreasing sequence.

\(X_d\) :

Certain zero-dimensional tropical variety.

\({G}_d\) :

Certain one-dimensional tropical variety.

\(\mathbf C, \mathbf b\) :

Cayley matrix and lifting vector.

q :

Last polytope so that \({m_{q}^{(d)}}\) increased at time d.

\(\Delta _j\) :

Pivoting direction, in \(\xi \)-space.

\(I_j\) :

Pivoting distance.

\(\Delta _j \varvec{\xi }, \Delta _j \varvec{\lambda }\) :

Pivoting vectors while dropping constraint j.

\(\mathbf {C}_{\mathrm {act}}, \mathbf b_{\mathrm {act}}\) :

Matrix and vector of active constraints.

\(\mathbf {C}_{\mathrm {inact}}, \mathbf b_{\mathrm {inact}}\) :

Matrix and vector of inactive constraints.

\(t(i,\mathbf a)\) :

score of inactive constraint \([i, \mathbf a]\)

\(Q_i\) :

Unit vector orthogonal to \(F_{i-1}\) in \(F_i\).

R :

nonstandard number, \(R>k\) for all \(k \in \mathbb R\).

B :

Inverse to the matrix \(\mathbf {C}_{\mathrm {act}}\) of active constraints.

\(t_{i,a}(R)\) :

Scores for inactive constraints.

\(\Gamma = (\mathcal V, \mathcal E)\) :

Graph to be explored. Union of tropical curves.

\(v_d\) :

number of vertices of \({G}_d\).

\(E_i\) :

Degree of 1-skeleton of lifting of \(A_i\)

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Authors and Affiliations

  1. Departamento de Matemática Aplicada, Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, Rio de Janeiro, RJ, 21941-909, Brazil

    Gregorio Malajovich

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Correspondence to Gregorio Malajovich.

Additional information

Communicated by Bernd Sturmfels.

A substantial part of this paper was written while visiting the Simons Institute for the Theory of Computing in the University of California at Berkeley. This visit was funded by CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior, Brazil. Proc. BEX 2388/14-6). This research is also funded by CNPq, Grants 441678/2014-9 and 306673/2013-4. Numerical experiments were performed at NACAD (Núcleo Avançado de Computação de Alto Desempenho) at UFRJ.

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Malajovich, G. Computing Mixed Volume and All Mixed Cells in Quermassintegral Time. Found Comput Math 17, 1293–1334 (2017). https://doi.org/10.1007/s10208-016-9320-1

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