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**A canonical polynomial for the volume of a polyhedron.**
*(English.
Russian original)*
Zbl 0941.52009

Russ. Math. Surv. 54, No. 2, 430-431 (1999); translation from Usp. Mat. Nauk 54, No. 2, 165-166 (1999).

Let there be given a polyhedron in \(\mathbb R^3,\) homeomorphic to the sphere, with triangular faces and \(n\) vertices, numbered from 1 to n in some order. Let \(x =\{x_1,y_1,z_1,x_2,y_2,z_2,\dots,x_n,y_n,z_n\}\) be the coordinates of the vertices, written out in numerical order, and let \(l=\{l_1,\dots ,l_e\}\) be the squares of the lengths of the edges, numbered in some order from 1 to \(e = 3n - 6\); let \(V\) be the volume, and let \(l(x)\) and \(V(x)\) the expressions for \(l\) and \(V\) in terms of the coordinates.

It is known [I. Kh. Sabitov, Fundam. Prikl. Mat. 2, No. 4, 1235-1246 (1996; Zbl 0904.52002) (Russian)], that for any polyhedron in \(\mathbb R^3\) there is a polynomial \(P_0 \in \mathbb Q[l][V]\) such that the leading coefficient LC\((P_0) = 1\) (that is, \(P_0\) has the form \(V^N+ \sim_{n<N} p_n(l)V^n,\) where the \(p_n(l)\) are polynomials in \(l\)) and \(P_0(l(x), V(x)) = 0.\) In particular, this polynomial enables us, in terms of the known metric of the polyhedron, to obtain a finite set of numbers, the roots of the polynomial, that contains the values of the volumes of all realizations (including the complex ones) of the metric of this polyhedron. But the algorithm for constructing this polynomial does not produce a unique result and many such polynomials can be constructed by it; also, it certainly gives a polynomial of much higher degree than is actually necessary. The authors give a unique method for constructing the required polynomial of minimum degree.

Let \(S\) be the set of polynomials \(Q \in \mathbb Q[l][V]\) such that \(Q(l(x), V(x)) = 0,\) not necessarily having leading coefficient 1. The following theorem gives some properties of this set. Let \(d\) be the smallest (non-zero) degree of the polynomials in \(S.\) Then among the polynomials of degree \(d\) in \(S\) there is a polynomial \(Q_0\) with LC\((Q_0) = 1.\) The polynomial \(Q_0\) divides all the polynomials in \(S.\) Such a polynomial \(Q_0\) is unique.

It is known [I. Kh. Sabitov, Fundam. Prikl. Mat. 2, No. 4, 1235-1246 (1996; Zbl 0904.52002) (Russian)], that for any polyhedron in \(\mathbb R^3\) there is a polynomial \(P_0 \in \mathbb Q[l][V]\) such that the leading coefficient LC\((P_0) = 1\) (that is, \(P_0\) has the form \(V^N+ \sim_{n<N} p_n(l)V^n,\) where the \(p_n(l)\) are polynomials in \(l\)) and \(P_0(l(x), V(x)) = 0.\) In particular, this polynomial enables us, in terms of the known metric of the polyhedron, to obtain a finite set of numbers, the roots of the polynomial, that contains the values of the volumes of all realizations (including the complex ones) of the metric of this polyhedron. But the algorithm for constructing this polynomial does not produce a unique result and many such polynomials can be constructed by it; also, it certainly gives a polynomial of much higher degree than is actually necessary. The authors give a unique method for constructing the required polynomial of minimum degree.

Let \(S\) be the set of polynomials \(Q \in \mathbb Q[l][V]\) such that \(Q(l(x), V(x)) = 0,\) not necessarily having leading coefficient 1. The following theorem gives some properties of this set. Let \(d\) be the smallest (non-zero) degree of the polynomials in \(S.\) Then among the polynomials of degree \(d\) in \(S\) there is a polynomial \(Q_0\) with LC\((Q_0) = 1.\) The polynomial \(Q_0\) divides all the polynomials in \(S.\) Such a polynomial \(Q_0\) is unique.

Reviewer: Serguey M.Pokas (Odessa)