## 6 Propagation Time for a Signal, Emitted and Intercepted by Satellites on a Space-Oriented Orbit
### Second Formulation of the Theorem for Four-Dimensional Uniformization
#### 12.6.4 Second Way for Representation of \(m^{2}\) and of \((\rho^{\prime}(z_{1}))^{2}\)  
The second way is based on the standard formulae for the parametrizable form of a cubic curve:  
\[(\rho^{\prime}(z_{1}))^{2}=4\rho^{3}(z_{1})-g_{2}\rho(z_{1})-g_{3}. \tag{290}\]  
Substituting into (284) for \(m^{2}\), we obtain the second representation:  
\[m^{2}=-2+\frac{g_{2}}{2}.\frac{1}{\rho^{2}(z_{1})}+\frac{g_{3}}{2}\frac{1}{ \rho^{3}(z_{1})}. \tag{291}\]  
Next we shall use also expression (258) for \(g_{3}\) through \(g_{2}\):  
\[g_{3}=g_{2}N(q),\,N(q):=\left(\frac{\overline{g}_{3}}{\overline{g}_{2}}\right) ^{\frac{81}{66}}(9K(q))^{\frac{25}{36}}.\]  
Taking this equality into account and also that according to (260), \(N(q)=8F(q)\), the second representation (291) for \(m^{2}\) can be written also in the form  
\[m^{2}=\frac{-\rho^{3}(z_{1})+\left[\frac{\rho(z_{1})}{2}+4F(q)\right]g_{2}}{ \rho^{3}(z_{1})}. \tag{292}\]  
Setting up equal the two representations (288) and (291) for \(m^{2}\), one can obtain  
\[-34\rho^{3}(z_{1})-3\rho^{3}(z_{1})\sqrt{12^{2}-\frac{2}{\rho^{6}(z_{1})}}\]  
\[=\frac{\delta z}{2}(\rho(z_{1})+8F(q)). \tag{293}\]  
6.5 Another Interpretation of the Equation (293) for the Equality of the Two Representations of \(m^{2}\)--A Theorem from Higher Algebra for Investigating Whether There are Roots of an Algebraic Polynomial  
After some transformations, the above equation can be represented in the form of a sixth-order algebraic equation with respect to \(\rho(z_{1})\):  
\[\left[9.12^{2}\right]\rho^{6}(z_{1})-34g_{2}\rho^{4}(z_{1})-68.4F(q)\rho^{3}( z_{1})\]  
\[-g_{2}^{2}\Big{[}\rho^{2}(z_{1})-4F(q)\rho(z_{1})+16F^{2}(q)\Big{]}=0. \tag{294}\]  
This equation of the 6 th degree is interesting, because it does not always have roots. It will, however, have roots for certain values of the invariant \(g_{2}\). Thus, the values of \(\rho(z_{1})\) are interrelated to the values of \(g_{2}\). It is interesting to investigate the following problem: According to (293), for a given function \(\rho(z_{1})\), there should be only one value of \(g_{2}\). However, the algebraic Equation (294), if solved as a quadratic equation with respect to \(g_{2}\) and for a given function \(\rho(z_{1})\), should have two roots with respect to \(g_{2}\), which will contradict the unique expression of \(g_{2}\) from (293). Now, let us give the formulation of the theorem of Schur from higher algebra, which gives the possibility to establish whether a polynomial has roots. The formulation of the theorem, originally published by Schur [62] in 1918, is taken from the monograph [61], where many other theorems concerning roots of polynomials can be found. The theorem of Schur has been applied in the paper [20] for establishing whether two different algebraic equations of the fourth degree have roots. The first of these equations was related to the introduced in [20] physical notion of a \"space-time interval\", which, for a particular case, was proved to be positive, negative or even equal to zero. By means of the Schur theorem, it was proved that the corresponding algebraic equation has roots. The second equation was related to the notion of \"geodesic distance\", which is related to the propagation of a light or radio signal in a gravitational field and, therefore, should be positive. Respectively, again, by applying the Schur theorem, it was proved that the corresponding algebraic equation does not have roots. The confirmation of the physical definitions has been commented on in the paper [40] and also the review paper [22]. This consistency between the mathematical results and the corresponding physical definitions give some grounds to believe that the Schur theorem is a reliable mathematical tool for the investigation, also, of Equation (294). **Theorem 13** ([62]). : _The necessary and sufficient conditions for the polynomial of \(n\)-th degree_  
\[f(y)=a_{0}y^{n}+a_{1}y^{n-1}+....+a_{n-2}y^{2}+a_{n-1}y+a_{n} \tag{295}\]  
_to have roots only in the circle \(\mid y\mid<1\) are the following ones:_  
1.