Complete Singularity Analysis for the Perspective-Four-Point Problem

Abstract

This paper is concerned with pose estimation and visual servoing from four points. We determine the configurations for which the corresponding Jacobian matrix becomes singular, leading to inaccurate and unstable results. Using an adequate representation and algebraic geometry, it is shown that, for any orientation between the camera and the object, there are always two to six singular locations of the camera in the generic case where the points are not coplanar, corresponding to the intersection of four cylinders. The particular case where the four points are coplanar is also characterized. Furthermore, some realistic example configurations are considered to substantiate the theory and to demonstrate failure cases in pose estimation and image-based visual servoing when the camera approaches a singularity.

Appendices

Appendices

1.1 A. Cylinders Equations

Equations of the four cylinders $C_{ijk}$ containing points $M_i,M_j,M_k$:

$$\begin{aligned} C_{123}&=-{X}^{2}d_{{2}}+Xd_{{2}}-{Y}^{2}d_{{2}}+ \left( {d_{{1}}}^{2} +{d_{{2}}}^{2}-d_{{1}} \right) Y, \end{aligned}$$

(47)

$$\begin{aligned} C_{124}&= \left( -{d_{{4}}}^{2}-{d_{{5}}}^{2} \right) {X}^{2}\nonumber \\&\quad +\left( {d_{{4}}}^{2}+{d_{{5}}}^{2} \right) X-{Y}^{2}{d_{{4}}}^{2} -2\,YZd_{{4}}d_{{5}} \nonumber \\&\quad +d_{{4}} \left( {d_{{3}}}^{2}+{d_{{4}}}^{2}+{d_{{5}}}^{2}-d_{{3}}\right) Y\nonumber \\&\quad -{Z}^{2}{d_{{5}}}^{2}+d_{{5}} \left( {d_{{3}}}^{2} +{d_{{4}}}^{2}+{d_{{5}}}^{2}-d_{{3}} \right) Z, \end{aligned}$$

(48)

$$\begin{aligned} C_{134}&= \left( -{d_{{1}}}^{2}{d_{{4}}}^{2}-{d_{{1}}}^{2}{d_{{5}}}^{2} +2\,d_{{1}}d_{{2}}d_{{3}}d_{{4}}-{d_{{2}}}^{2}{d_{{3}}}^{2} \right) {X}^{2}\nonumber \\&\quad -2\,d_{{1}}d_{{2}}{d_{{5}}}^{2}XY+2\,d_{{2}}d_{{5}} \left( d_{{1}}d_{{4} }-d_{{2}}d_{{3}} \right) ZX\nonumber \\&\quad + \left( {d_{{1}}}^{3}{d_{{4}}}^{2}+{d_{{1}}}^{3}{d_{{5}}}^{2} -{d_{{1}}}^{2}d_{{2}}d_{{3}}d_{{4}}+d_{{1}}{d_{{2}}}^{2}{d_{{4}}}^{2}\right. \nonumber \\&\quad +d_{{1}}{d_{{2}}}^{2}{d_{{5}}}^{2}-d_{{1}}d_{{2}}{d_{{3}}}^{2}d_{{4}} -d_{{1}}d_{{2}}{d_{{4}}}^{3}-d_{{1}}d_{{2}}d_{{4}}{d_{{5}}}^{2}\nonumber \\&\quad \left. -{d_{{2}}}^{3}d_{{3}}d_{{4}}+{d_{{2}}}^{2}{d_{{3}}}^{3} +{d_{{2}}}^{2}d_{{3}}{d_{{4}}}^{2}+{d_{{2}}}^{2}d_{{3}}{d_{{5}}}^{2}\right) X\nonumber \\&\quad + \left( -{d_{{1}}}^{2}{d_{{4}}}^{2}+2\,d_{{1}}d_{{2}}d_{{3}}d_{{4}} -{d_{{2}}}^{2}{d_{{3}}}^{2}-{d_{{2}}}^{2}{d_{{5}}}^{2}\right) {Y}^{2}\nonumber \\&\quad -2\,d_{{1}}d_{{5}} \left( d_{{1}}d_{{4}}-d_{{2}}d_{{3}} \right) ZY\nonumber \\&\quad + \left( -{d_{{1}}}^{3}d_{{3}}d_{{4}}+{d_{{1}}}^{2}d_{{2}} {d_{{3}}}^{2}+{d_{{1}}}^{2}d_{{2}}{d_{{5}}}^{2}+{d_{{1}}}^{2}{d_{{3}}} ^{2}d_{{4}}\right. \nonumber \\&\quad +{d_{{1}}}^{2}{d_{{4}}}^{3} +{d_{{1}}}^{2}d_{{4}}{d_{{5}}}^{2} -d_{{1}}{d_{{2}}}^{2}d_{{3}}d_{{4}} -d_{{1}}d_{{2}}{d_{{3}}}^{3}\nonumber \\&\quad \left. -d_{{1}}d_{{2}}d_{{3}}{d_{{4}}}^{2}-d_{{1}}d_{{2}}d_{{3}}{d_{{5}}}^{2} +{d_{{2}}}^{3}{d_{{3}}}^{2}+{d_{{2}}}^{3}{d_{{5}}}^{2} \right) Y\nonumber \\&\quad -{d_{{5}}}^{2} \left( {d_{{1}}}^{2}+{d_{{2}}}^{2} \right) {Z}^{2}\nonumber \\&\quad -d_{{5}} \left( {d_{{1}}}^{2}+{d_{{2}}}^{2} \right) \left( d_{{1}}d_{{3}}+d_{{2}}d_{{4}}-{d_{{3}}}^{2} -{d_{{4}}}^{2}-{d_{{5}}}^{2} \right) Z, \end{aligned}$$

(49)

$$\begin{aligned} C_{234}&=-d_{{5}} \left( {d_{{1}}}^{3}d_{{3}}+{d_{{1}}}^{2}d_{{2}}d_{{4}} -{d_{{1}}}^{2}{d_{{3}}}^{2}-{d_{{1}}}^{2}{d_{{4}}}^{2}\right. \nonumber \\&\quad -{d_{{1}}}^{2}{d_{{5}}}^{2}+d_{{1}}{d_{{2}}}^{2}d_{{3}} +{d_{{2}}}^{3}d_{{4}}-{d_{{2}}}^{2}{d_{{3}}}^{2}\nonumber \\&\quad -{d_{{2}}}^{2}{d_{{4}}}^{2}-{d_{{2}}}^{2}{d_{{5}}}^{2}-{d_{{1}}}^{3} -{d_{{1}}}^{2}d_{{3}}-d_{{1}}{d_{{2}}}^{2}+2\,d_{{1}}{d_{{3}}}^{2}\nonumber \\&\quad +2\,d_{{1}}{d_{{4}}}^{2}+2\,d_{{1}}{d_{{5}}}^{2}-{d_{{2}}}^{2}d_{{3}}\nonumber \\&\quad +2\,{d_{{1}}}^{2}-d_{{1}}d_{{3}}+2\,{d_{{2}}}^{2}-d_{{2}}d_{{4}}\nonumber \\&\quad \left. -{d_{{3}}}^{2}-{d_{{4}}}^{2}-{d_{{5}}}^{2}-d_{{1}}+d_{{3}} \right) Z\nonumber \\&\quad +\left( -{d_{{1}}}^{3}d_{{3}}d_{{4}}+{d_{{1}}}^{2}d_{{2}}{d_{{3}}}^{2} +{d_{{1}}}^{2}d_{{2}}{d_{{5}}}^{2}+{d_{{1}}}^{2}{d_{{3}}}^{2}d_{{4}}\right. \nonumber \\&\quad +{d_{{1}}}^{2}{d_{{4}}}^{3}+{d_{{1}}}^{2}d_{{4}}{d_{{5}}}^{2} -d_{{1}}{d_{{2}}}^{2}d_{{3}}d_{{4}}-d_{{1}}d_{{2}}{d_{{3}}}^{3}\nonumber \\&\quad -d_{{1}}d_{{2}}d_{{3}}{d_{{4}}}^{2}-d_{{1}}d_{{2}}d_{{3}}{d_{{5}}}^{2} +{d_{{2}}}^{3}{d_{{3}}}^{2}+{d_{{2}}}^{3}{d_{{5}}}^{2}+{d_{{1}}}^{3}d_{{4}}\nonumber \\&\quad -2\,{d_{{1}}}^{2}d_{{2}}d_{{3}}+{d_{{1}}}^{2}d_{{3}}d_{{4}} +d_{{1}}{d_{{2}}}^{2}d_{{4}}+d_{{1}}d_{{2}}{d_{{3}}}^{2}\nonumber \\&\quad +d_{{1}}d_{{2}}{d_{{4}}}^{2}+d_{{1}}d_{{2}}{d_{{5}}}^{2} -2\,d_{{1}}{d_{{3}}}^{2}d_{{4}}-2\,d_{{1}}{d_{{4}}}^{3}-2\,d_{{1}}d_{{4}}{d_{{5}}}^{2}\nonumber \\&\quad -2\,{d_{{2}}}^{3}d_{{3}}+{d_{{2}}}^{2}d_{{3}}d_{{4}} +d_{{2}}{d_{{3}}}^{3}+d_{{2}}d_{{3}}{d_{{4}}}^{2}+d_{{2}}d_{{3}}{d_{{5}}}^{2}\nonumber \\&\quad +{d_{{1}}}^{2}d_{{2}}-2\,{d_{{1}}}^{2}d_{{4}}+d_{{1}}d_{{2}}d_{{3}}\nonumber \\&\quad +d_{{1}}d_{{3}}d_{{4}}+{d_{{2}}}^{3}-{d_{{2}}}^{2}d_{{4}}-2\,d_{{2}}{d_{{3}}}^{2}\nonumber \\&\quad -d_{{2}}{d_{{4}}}^{2}-2\,d_{{2}}{d_{{5}}}^{2}+{d_{{3}}}^{2}d_{{4}}\nonumber \\&\quad +{d_{{4}}}^{3}+d_{{4}}{d_{{5}}}^{2}-d_{{1}}d_{{2}}+d_{{1}}d_{{4}}+d_{{2}}d_{{3}}\nonumber \\&\quad \left. -d_{{3}}d_{{4}} \right) Y-{d_{{5}}}^{2} \left( {d_{{1}}}^{2}+{d_{{2}}}^{2}-2\,d_{{1}}+1 \right) {Z}^{2}\nonumber \\&\quad + \left( {d_{{1}}}^{3}{d_{{4}}}^{2}+{d_{{1}}}^{3}{d_{{5}}}^{2} -{d_{{1}}}^{2}d_{{2}}d_{{3}}d_{{4}}+d_{{1}}{d_{{2}}}^{2}{d_{{4}}}^{2}\right. \nonumber \\&\quad +d_{{1}}{d_{{2}}}^{2}{d_{{5}}}^{2}-d_{{1}}d_{{2}}{d_{{3}}}^{2}d_{{4}} -d_{{1}}d_{{2}}{d_{{4}}}^{3}-d_{{1}}d_{{2}}d_{{4}}{d_{{5}}}^{2}\nonumber \\&\quad -{d_{{2}}}^{3}d_{{3}}d_{{4}}+{d_{{2}}}^{2}{d_{{3}}}^{3} +{d_{{2}}}^{2}d_{{3}}{d_{{4}}}^{2}+{d_{{2}}}^{2}d_{{3}}{d_{{5}}}^{2}\nonumber \\&\quad +{d_{{1}}}^{2}d_{{2}}d_{{4}}-{d_{{1}}}^{2}{d_{{4}}}^{2} -{d_{{1}}}^{2}{d_{{5}}}^{2}+{d_{{2}}}^{3}d_{{4}}-{d_{{2}}}^{2}{d_{{3}}}^{2}\nonumber \\&\quad -2\,{d_{{2}}}^{2}{d_{{4}}}^{2}-2\,{d_{{2}}}^{2}{d_{{5}}}^{2} +d_{{2}}{d_{{3}}}^{2}d_{{4}}+d_{{2}}{d_{{4}}}^{3}+d_{{2}}d_{{4}}{d_{{5}}}^{2}\nonumber \\&\quad +d_{{1}}d_{{2}}d_{{4}}-d_{{1}}{d_{{4}}}^{2}-d_{{1}}{d_{{5}}}^{2} -{d_{{2}}}^{2}d_{{3}}+d_{{2}}d_{{3}}d_{{4}}+{d_{{2}}}^{2}\nonumber \\&\quad \left. -2\,d_{{2}}d_{{4}} +{d_{{4}}}^{2}+{d_{{5}}}^{2} \right) X\nonumber \\&\quad + \left( -{d_{{1}}}^{2}{d_{{4}}}^{2}+2\,d_{{1}}d_{{2}}d_{{3}}d_{{4}} -{d_{{2}}}^{2}{d_{{3}}}^{2}-{d_{{2}}}^{2}{d_{{5}}}^{2}\right. \nonumber \\&\quad -2\,d_{{1}}d_{{2}}d_{{4}}+2\,d_{{1}}{d_{{4}}}^{2}+2\,{d_{{2}}}^{2}d_{{3}}\nonumber \\&\quad \left. -2\,d_{{2}}d_{{3}}d_{{4}}-{d_{{2}}}^{2}+2\,d_{{2}}d_{{4}}-{d_{{4}}}^{2} \right) {Y}^{2}\nonumber \\&\quad -d_{{2}}{d_{{3}}}^{2}d_{{4}}-d_{{2}}d_{{4}}{d_{{5}}}^{2}+d_{{2}}d_{{3}}d_{{4}} + \left( -{d_{{1}}}^{2}{d_{{4}}}^{2}-{d_{{1}}}^{2}{d_{{5}}}^{2}\right. \nonumber \\&\quad +2\,d_{{1}}d_{{2}}d_{{3}}d_{{4}}-{d_{{2}}}^{2}{d_{{3}}}^{2} -2\,d_{{1}}d_{{2}}d_{{4}}+2\,d_{{1}}{d_{{4}}}^{2}+2\,d_{{1}}{d_{{5}}}^{2}\nonumber \\&\quad \left. +2\,{d_{{2}}}^{2}d_{{3}}-2\,d_{{2}}d_{{3}}d_{{4}}-{d_{{2}}}^{2} +2\,d_{{2}}d_{{4}}-{d_{{4}}}^{2}-{d_{{5}}}^{2} \right) {X}^{2}\nonumber \\&\quad -2\,d_{{2}}{d_{{5}}}^{2} \left( d_{{1}}-1 \right) XY\nonumber \\&\quad +2\,d_{{2}}d_{{5}} \left( d_{{1}}d_{{4}} -d_{{2}}d_{{3}}+d_{{2}}-d_{{4}} \right) ZX\nonumber \\&\quad -2\,d_{{5}} \left( d_{{1}}-1\right) \left( d_{{1}}d_{{4}}-d_{{2}}d_{{3}}+d_{{2}}-d_{{4}}\right) ZY\nonumber \\&\quad -d_{{2}}{d_{{4}}}^{3}-d_{{1}}{d_{{4}}}^{2}-d_{{1}}{d_{{5}}}^{2} -{d_{{2}}}^{2}d_{{3}}-{d_{{2}}}^{3}d_{{4}}+2\,{d_{{2}}}^{2}{d_{{4}}}^{2}\nonumber \\&\quad -{d_{{1}}}^{2}d_{{2}}d_{{4}}+d_{{1}}d_{{2}}d_{{4}} -d_{{1}}{d_{{2}}}^{2}{d_{{5}}}^{2}+{d_{{2}}}^{3}d_{{3}}d_{{4}}+d_{{1}}d_{{2}}{d_{{4}}}^{3}\nonumber \\&\quad -d_{{1}}{d_{{2}}}^{2}{d_{{4}}}^{2}-{d_{{2}}}^{2}d_{{3}}{d_{{4}}}^{2} -{d_{{2}}}^{2}d_{{3}}{d_{{5}}}^{2}-2\,d_{{1}}d_{{2}}d_{{3}}d_{{4}}\nonumber \\&\quad +{d_{{1}}}^{2}d_{{2}}d_{{3}}d_{{4}}+d_{{1}}d_{{2}}{d_{{3}}}^{2}d_{{4}} +d_{{1}}d_{{2}}d_{{4}}{d_{{5}}}^{2}+2\,{d_{{2}}}^{2}{d_{{5}}}^{2}\nonumber \\&\quad -{d_{{1}}}^{3}{d_{{4}}}^{2}-{d_{{1}}}^{3}{d_{{5}}}^{2}-{d_{{2}}}^{2}{d_{{3}}}^{3} +2\,{d_{{1}}}^{2}{d_{{4}}}^{2}+2\,{d_{{1}}}^{2}{d_{{5}}}^{2}\nonumber \\&\quad +2\,{d_{{2}}}^{2}{d_{{3}}}^{2} \end{aligned}$$

(50)

1.2 B. Algebraic Geometry Tools

Most of the following definitions and examples are taken from Cox et al. (2007).

1.2.1 Ideal

A subset ${\mathcal {I}}$ of the ring of polynomials in the variables $x_1,\ldots ,x_n$ with coefficients in a field k, $k[x_1, . . . , x_n]$ is an ideal if it satisfies:

$$\begin{aligned} \begin{array}{rl} (i)\, &{} \, 0 \in {\mathcal {I}}\\ (ii)\, &{} \, \text {If} \, f,g \in {\mathcal {I}} \, \text {then} \, f+g \in {\mathcal {I}}\\ (iii)\, &{} \, \text {If} \, f \in {\mathcal {I}} \, \text {and} \, h \in k[x_1, \ldots , x_n], \, \text {then} \, hf \in {\mathcal {I}}. \end{array} \end{aligned}$$

(51)

Let $f_1,\ldots ,f_s$ be polynomials in $k[x_1, \ldots , x_n]$, then

$$\begin{aligned} \langle f_1,\ldots ,f_s \rangle = \Bigg \{ \sum _{i=1}^{s} h_i f_i : h_1,\ldots ,h_s \in k[x_1, \ldots , x_n] \Bigg \} \end{aligned}$$

is an ideal generated by $f_1,\ldots ,f_s$. This set is interesting because, for any field K containing k, all polynomials $f\in \langle f_1,\ldots ,f_s \rangle$ give polynomial equations $f=0$ that hold at all points of $K^n$ at which $f_1=0,\ldots ,f_s=0$ hold simultaneously, plus it has a structure that allows many useful algebraic operations.

1.2.2 Variety

If $f_1,\ldots ,f_s$ are polynomials in the ring $k[x_1,\ldots ,x_n]$ and if

$$\begin{aligned} {V}(f_1,\ldots ,f_s)=\{ (a_1,\ldots ,a_n) \in k^n : f_i(a_1,\ldots ,a_n)=0, \, \text {for all} \, 1 \le i \le s \} \end{aligned}$$

then ${V}(f_1,\ldots ,f_s)$ is called an affine variety defined by the polynomials $f_i$ (note that in many other text books, affine varieties are defined the same way but when k is algebraically closed, e.g. $k={\mathbb {C}}$). It is the set of all solutions with coordinates in k of the system of equations $f_1(x_1, \ldots , x_n) =\cdots = f_s (x_1, \ldots , x_n) = 0$.

There is a dual relation between certain varieties and certain ideals (see Cox et al. (2007) for the precise conditions on them):

$$\begin{aligned} \mathrm {varieties}&\longleftrightarrow \mathrm {ideals}\\ V=V(I)&\longleftrightarrow I=I(V) , \end{aligned}$$

where V(I) is the set of all points in $k^n$ at which all polynomials $f\in {\mathbb {I}}$ vanish, and I(V) is the set of all polynomials in $k[x_1,\ldots ,x_n]$ vanishing at all points in V.

If I and J are two ideals in $k[x_1,\ldots ,x_n]$ and $I\subset J$ (that is, J contains more polynomials than I), then $V(J)\subset V(I)$. This is because the points where all polynomials in J vanish make also all the polynomials in I vanish, but the additional polynomials in J impose additional conditions, so not all points in V(I) are necessarily in V(J) (more equations mean less solutions).

1.2.3 Difference of Varieties Quotient of Ideals

The set difference of two affine varieties is generally not an affine variety. For instance in ${\mathbb {C}}^2$, the difference $V(\langle x \rangle ) \setminus V(\langle y \rangle )$ is the set resulting from removing the x-axis from the y-axis. This leaves the y-axis minus the point $\{0,0\}$, which is not a variety but an open subset of a variety. It cannot be written as the set of solutions of a system of polynomial equations (it is not an affine variety). The smallest affine variety which contains it, is called the Zariski closure of the difference, denoted with an overline. Loosely speaking, taking the Zariski closure amounts to patching up the holes in the open set. Therefore, in this example, the Zariski closure of the difference is the y-axis itself:

$$\begin{aligned} \overline{V(\langle x \rangle ) \setminus V(\langle y \rangle )}=V(\langle x \rangle ). \end{aligned}$$

Using the correspondence between ideals and varieties, an ideal defining $\overline{V \setminus W}$ where V and W are affine varieties is obtained as the saturation of an ideal I defining V with an ideal J defining W. It is denoted by $I:J^\infty$. The colon symbolizes the quotient of one ideal w.r.t. the other, and it removes factors from the polynomials in the first ideal which appear as polynomials themselves in the second ideal. The infinity symbol corresponds to considering all products of arbitrarily many polynomials in the second ideal. A couple of examples:

$$\begin{aligned}&\langle x \rangle : \langle y\rangle ^\infty =\langle x \rangle ,\\&\langle xy^2 \rangle : \langle y\rangle =\langle xy\rangle \subset \langle xy^2 \rangle : \langle y^2\rangle =\langle xy^2 \rangle : \langle y\rangle ^\infty =\langle x \rangle . \end{aligned}$$

Then,

$$\begin{aligned} \langle x \rangle : \langle y \rangle ^\infty =\langle x \rangle .&\leftrightarrow \overline{V(\langle x \rangle ) \setminus V (\langle y \rangle )}=V(\langle x \rangle ) ,\\ \langle xy^2 \rangle : \langle y \rangle ^\infty =\langle x \rangle .&\leftrightarrow \overline{V(\langle xy \rangle ) \setminus V (\langle y \rangle )}=V(\langle x \rangle ). \end{aligned}$$

In the context of this paper, to find out whether actually $V(I_{4})=V(I_{28})$ or the first one strictly contains the second one, we are interested in computing the Zariski closure of the difference $\overline{V(I_4)\setminus V(I_{28})}$. Dually, determining this set amounts to computing the saturation of the ideal $I_4$ w.r.t. the ideal $I_{28}$, denoted by $I_4:I_{28}^\infty$. This is equivalent to removing from the polynomials in $I_4$ all the factors which are themselves polynomials in $I_{28}$. It can be done in a computational environment such as Maple or Singular and it results in (25), where

$$\begin{aligned} I_d=I_4:I_{28}^\infty ,\; \text {and}\;V(I_d)=\overline{V(I_4)\setminus V(I_{28})}. \end{aligned}$$

1.2.4 Gröbner Basis

We can find different sets of polynomial equations having the same solution set. An especially useful one is the Gröbner basis. A Gröbner Basis of an ideal I can be defined as a finite set generating the ideal and such that the leading terms of the elements in this set (according to some prescribed ordering of the monomials⁶) generate the leading terms of all elements in the ideal I. One of its many uses is to solve polynomial equations. To do so, it requires an ordering of the terms in the polynomials. The ordering used in this paper is lexicographic ordering defined formally as follows:

Let $\alpha = (\alpha _1, \ldots , \alpha _n)$ and $\beta = (\beta _1, \ldots , \beta _n) \in {\mathbb {Z}}_{\ge 0}^{n}$. It is said that $\alpha >_{lex} \beta$ if, in the vector difference $\alpha -\beta \in n$, the leftmost nonzero entry is positive. The monomials $x^{\alpha } >_{lex} x^{\beta }$ if $\alpha >_{lex} \beta$.

1.3 C. Validity of Gröbner Basis of the Cylinder Polynomials

The fact that the set of polynomials $\left\{ F_1,F_2,F_3\right\}$ in (29-31) is a Gröbner basis of the ideal $\langle C_{123},C_{124},C_{134},C_{234} \rangle$ implies that the polynomial system $\left\{ F_1=F_2=F_3=0\right\}$ has the same solutions as the polynomial system $\left\{ C_{123}=C_{124}=\right.$$\left. C_{134}=C_{234} =0\right\}$.

When one has, as in this case, two sets of generators for an ideal, then one can write the polynomials in the first set as a linear combination of the polynomials in the second set with coefficients in the real numbers and the parameters, and vice versa. On one side, the fact that the set $\left\{ F_1,F_2,F_3\right\}$ can be expressed as a combination of the set $\left\{ C_{123},C_{124}, C_{134},C_{234} \right\}$ means that for $i=1,2,3$:

$$\begin{aligned} F_i=\frac{\alpha _i}{\alpha '_i}C_{123}+\frac{\beta _i}{\beta '_i}C_{124} +\frac{\gamma _i}{\gamma '_i}C_{134}+ \frac{\delta _i}{\delta '_i}C_{234}, \end{aligned}$$

(52)

where $\alpha _i,\beta _i,\gamma _i,\delta _i,\alpha '_i,\beta '_i,\gamma '_i,\delta '_i$ are some polynomials in the variables $d_1,\ldots ,d_5$ with real coefficients. However, we have computed a Gröbner basis with parameters in the coefficients, and this means that, for some choices of the parameter values, the denominators $\alpha '_i$, $\beta '_i$, $\gamma '_i$, $\delta '_i$ could be zero. For such values of the parameters, the Gröbner basis we have obtained will not be valid. Similarly, for $i,j,k\in \{ 1,2,3,4\}$,

$$\begin{aligned} C_{ijk}=\frac{A_{ijk}}{A'_{ijk}}F_1 +\frac{B_{ijk}}{B'_{ijk}}F_2+\frac{C_{ijk}}{C'_{ijk}}F_3 \end{aligned}$$

(53)

with $A_{ijk},B_{ijk},C_{ijk},A'_{ijk},B'_{ijk}, C'_{ijk}\in {\mathbb {R}}[d_1,\ldots ,d_5]$. For the same reason as before, whenever the choice of the parameter values is such that $A'_{ijk}$, $B'_{ijk}$ or $C'_{ijk}$ vanish for some i, j, k, then $C_{ijk}$ cannot be written as a linear combination of $F_1,F_2,F_3$. Computing the coefficients of these combinations, we come to the conclusion that the Gröbner basis is valid as long as the following polynomial given by the least common multiple of the denominators in (52) and (53) does not vanish:

$$\begin{aligned} q=\mathrm {lcm}\left( \alpha '_1,\beta '_i,\gamma '_i,\delta '_i, A'_{ijk},B'_{ijk},C'_{ijk} \right) ,\;i,j,k\in \{ 1,2,3,4\}. \end{aligned}$$

(54)

In fact, the above calculations can be avoided due to the fact that the product $c_6\ c_y\ c_z$ (see 29–31) contains all the factors of q (except the trivial factors corresponding to the coincidence of some $M_i$ and $M_j,i \ne j =1,2,3,4$) and hence for any parameter choice for which $c_6\ c_y\ c_z=0$, the calculated Gröbner basis is not valid (see the supplementary Maple file). We refer to Kalkbrener (1997) for more involved results on the stability of Gröbner bases under specialization.