Reports of the Academy of Sciences of the USSR
1958. Volume 119, No. 1

MATHEMATICS

P. V. NIKOLAEV

ON THE UNIQUENESS OF NOMOGRAPHIC REPRESENTATIONS OF EQUATIONS

(Presented by Academician A. N. Kolmogorov, 14 X 1957)

The problem of the uniqueness of the representation by net nomograms of an equation in three variables has been considered by many authors \(\left({}^{1-5}\right)\), etc., but has not yet been solved. Let us consider it for an equation, analytic in some domain \(G\), in three pairs of variables

\[ F(t,\tau)=F(t_1,\tau_1;\ t_2,\tau_2;\ t_3,\tau_3)=0, \tag{1} \]

admitting in \(G\) an anamorphosis

\[ \Phi(t,\tau)=\psi(t,\tau)\cdot F(t,\tau)= \left|f_{i1}(t_i,\tau_i);\ f_{i2}(t_i,\tau_i);\ f_{i3}(t_i,\tau_i)\right| \tag{2} \]

with a prescribed partition of the variables into pairs. Here \(\Phi(t,\tau)\) is also a function analytic in \(G\) \((\Phi \not\equiv 0)\);

\[ \psi(t,\tau)= \psi_1(t_2,\tau_2;\ t_3,\tau_3)\cdot \psi_2(t_3,\tau_3;\ t_1,\tau_1)\cdot \psi_3(t_1,\tau_1;\ t_2,\tau_2); \tag{3} \]

\(A\) is a multiplier of equation (1). We shall assume that \(F\) is a nondegenerate function, i.e., different from a function of the form (3). The equations \(F=0\) and \(\Phi=0\) will be called similar, in particular, \(N\)-equivalent, if \(\psi\) has the form \(\varphi_1(t_1,\tau_1)\cdot\varphi_2(t_2,\tau_2)\cdot\varphi_3(t_3,\tau_3)\).

In the case of an algebraic equation in three variables

\[ F(t)=F(t_1,t_2,t_3)=0 \tag{4} \]

the problem of anamorphosis, as follows from the lemmas on the rationality of anamorphoses \(\left({}^{6}\right)\), is also solved in the class of algebraic equations; moreover, the bases \(C_i\) and \(\overline C_i\) of the variable \(t_i\) in any two anamorphoses of equation (4) will be birationally equivalent. If, therefore, in an anamorphosis (1) (with \(\tau_i=\mathrm{const}\)) of equation (4) birational parameters have been introduced for the curves \(C_i\),

\[ s_i=f_i(t_i)\quad (i=1,2,3), \tag{5} \]

i.e., Lüroth parameters \(\left({}^{7}\right)\), then the \(s_i\) will be Lüroth parameters in every other anamorphosis of equation (4) as well. Thus the problem is reduced, by the change of variables (5) from \(t_i\) to \(s_i\), to the consideration, instead of (4), of a birational equation in the variables \(s_i\) similar to it, i.e., an equation in whose anamorphoses the points of the base \(C_i\) are birational functions of the parameter \(s_i\).

Lemma 1 (on the uniqueness of anamorphoses). Suppose that in each of two nomograms of some nondegenerate equation \(F(t)=0\), analytic in the domain \(G\), there exists the following configuration of points: three noncollinear points \(A_k\) \((k=1,2,3)\) with labels, respectively, \(t_3=t_{3k}\)

\((k=1,2,3)\), the line \(A_1A_3\) with the points \(t_i=t_{i1}\) \((i=1,2)\), and the line \(A_2A_3\) with the points \(t_i=t_{i2}\) \((i=1,2)\). If the points of the nomogram \(t_i=t_{i1}\) \((i=1,2)\), distinct from one another, and also the points \(t_i=t_{i2}\) \((i=1,2)\) (possibly coincident) are different from the vertices of the triangle \(A_1A_2A_3\), then these nomograms are projective.

Indeed, let (2) (for \(\tau_i=\mathrm{const}\)) and

\[ \bar{\Phi}|t|=\bar{\psi}(t)\cdot \Phi(t)=|\bar f_{i1}(t_i);\ \bar f_{i2}(t_i);\ \bar f_{i3}(t_i)| \tag{6} \]

be two such anamorphoses. Taking \(A_1A_2A_3\) as the coordinate triangle in (2) and (6), one may, with a proper choice of the unit point, write the expansion of the function \(\Phi(t)\) with respect to \(t_3\) in the form

\[ \bar{\Phi}(t)=\bar{\psi}_3(t_1,t_2)\cdot \sum_{k=1}^{3}\bar{\psi}_1^{(0k)}\bar{\psi}_2^{(k0)}\Phi^{(00k)}\bar f_{3k}, \tag{7} \]

where \(\Phi^{(00k)}=\Phi(t_1,t_2,t_{3k})\), \(\bar{\psi}_1^{(0k)}=\bar{\psi}_1(t_2,t_{3k})\), etc.

Applying to (7) the condition of an anamorphosis with a simple \(A\)-multiplier \((^8)\), we obtain the identities

\[ \sum_{k=1}^{3}\bar f_{ik}(t_i)\cdot \bar{\psi}_1^{(0k)}\bar{\psi}_2^{(0k)}\Phi^{(00k)} \equiv 0 \quad (i=1,2). \tag{8} \]

It can be shown that from (8), under the conditions of the lemma, the elements \(\bar f_{ik}\) \((i=1,2;\ k=1,2,3)\) are determined, in essence, uniquely; but then this will also be true for the elements \(\bar f_{3k}\) \((k=1,2,3)\). Hence it follows:

Theorem 1. All anamorphoses of an algebraic (irreducible) equation (4) which is not similar to an equation of the third \(N\)-order are projective.

Indeed, equation (4) may be regarded as a reduced equation, i.e., one free of divisors with two variables. If it admits only a simple \(A\)-multiplier, then the question is clear \((^8)\). If, however, it admits a general \(A\)-multiplier, then the order of the basis common to all variables will be greater than 3, and therefore any two nomograms of such an equation will have the configuration of points of Lemma 1; the theorem follows from this.

In the case where equation (1) is pseudoalgebraic \((^9)\) in three variables,

\[ F(t)=\sum_{i=0}^{m} a_i(t_2,t_3)\cdot t_1^i=0, \tag{9} \]

where the \(a_i\) are functions analytic in \(G\), it proves possible to consider it in a certain cylindrical domain \(H\)

\[ |t_1|<\infty;\qquad |t_i-\alpha_i|<h\quad (i=1,2,3), \tag{10} \]

where \(t_i=\alpha_i\) \((i=1,2,3)\) is the point \(\alpha\in G\). Namely, it is not difficult to show that similarity relations and \(N\)-equivalence of equations holding locally (in some subdomain \(E\subset G\)) will also hold “as a whole” (in the entire domain \(G\)). But from this, by virtue of the lemmas on the rationality of anamorphoses \((^6)\), it will follow that an anamorphosis constructed in \(E\subset G\) for the equation \(F(t)=0\) will, up to \(N\)-equivalence, represent it also in \(G\), or in \(H\), having a nonempty intersection with \(G\). It is also possible to show that, in a linear anamorphosis of equation (9) given in the domain \(H\), the basis \(C_2\) will then and only then lie on \(C_1\) if the pseudopolynomial \(F(t)\) (with vertex at \((\alpha_2,\alpha_3)\)) is algebraically reducible \((^9)\) and has a nontrivial algebraic divisor in \(t_1\) and \(t_2\). Hence it follows that, for equation (9), freed of algebraic divisors and admitting an \(A\)-multiplier with the variable \(t_1\), a change of variables of the form (5) is possible

algebraization, at least with respect to one of the variables \(t_2, t_3\). Therefore the following theorem is valid.

Theorem 2. All anamorphoses of a nondegenerate pseudoalgebraic equation (9) that is not similar to an equation of the third \(N\)-order are projective.

In the case of equation (1) in three pairs of variables, the following is valid:

Theorem 3. If a nondegenerate analytic equation (1), algebraic (or algebraizable) with respect to at least one variable, is not similar to an equation of the third \(N\)-order, then all its anamorphoses (2) with a prescribed partition of the variables into pairs \((t_i,\tau_i)\) are projective.

If one were to assume that such an equation has nonprojective anamorphoses, then it would be possible, by substitutions of the form \(\tau_i=f_i(t_i)\) \((i=1,2,3)\), where \(f_i\) are polynomials, to construct a pseudoalgebraic equation \(F(t)=0\) which, contrary to Theorem 2, would admit nonprojective anamorphoses.

In the general case of a nonalgebraizable equation (1), suppose that equation (1) has a direct anamorphosis

\[ F(t,\tau)=\left| f_{i1}(t_i,\tau_i);\ f_{i2}(t_i,\tau_i);\ f_{i3}(t_i,\tau_i)\right| \tag{11} \]

and a nonprojective anamorphosis to it

\[ \Phi(t,\tau)=\psi(t,\tau)\cdot F(t,\tau)=\left|\bar f_{i1}(t_i,\tau_i);\ \bar f_{i2}(t_i,\tau_i);\ \bar f_{i3}(t_i,\tau_i)\right|. \tag{12} \]

It can be shown that the dimensionality of any of the equations

\[ \Phi_i(t,\tau)=\Phi(t,\tau):\psi_i=0 \qquad (i=1,2,3) \tag{13} \]

with respect to each of the pairs of variables \((t_k,\tau_k)\) is equal to three, and therefore in the linear expansion

\[ \Phi_3(t,\tau)=\sum_{k=1}^{3}\psi_1^{(0k)}\psi_2^{(k0)}F^{(00k)}\bar f_{3k} \tag{14} \]

among the system of generators, with respect to \((t_1,\tau_1)\), of this function:

\[ \varphi_{11}=\psi_2^{(10)}f_{12},\ldots;\ \varphi_{16}=\psi_2^{(30)}f_{11};\ldots \tag{15} \]

three functions are linear combinations of the others; an analogous conclusion holds for the functions \(\varphi_{2k}=\varphi_{2k}(t_2,\tau_2)\).

Considering further the two possible cases: linear independence or dependence of the functions \(\varphi_{2k}\) \((k=1,2,3)\), we find that in each of them the anamorphosis condition for equation (1) leads to a relation of the third degree with respect to the functions \(f_{1k}\) \((k=1,2,3)\). Thus, in the first case, the \(A\)-matrix \(T_3^{(23)}\) of equation (14) has the form

\[ \begin{array}{c|ccc} T_3^{(23)} & \varphi_{21} & \varphi_{22} & \varphi_{23} \\ \hline \bar f_{31} & -\varphi_{12} & \varphi_{11} & 0 \\ \bar f_{32} & -\gamma_1\varphi_{13} & -\gamma_2\varphi_{13} & -\varphi_{14}-\gamma_3\varphi_{13} \\ \bar f_{33} & \alpha_1\varphi_{16}-\beta_1\varphi_{15} & \alpha_2\varphi_{16}-\beta_2\varphi_{15} & \alpha_3\varphi_{16}-\beta_3\varphi_{15} \end{array} \tag{16} \]

where \(\alpha,\beta,\gamma\) are certain constants, and the anamorphosis condition \(\left|T_3^{(23)}\right|=0\) \(^{(8)}\) leads to the aforementioned relation between the \(f_{1k}\). If this relation is an equation for \(f_{1k}\) (and not an absolute identity), then it is clear that the variables \((t_1,\tau_1)\) have a binary scale on a curve of the third order of genus one. If, however, this relation is an absolute identity,

then, for the functions \(\varphi_{1k}\), it turns out that the relation

\[ \varphi_{11}\varphi_{14}\varphi_{16}-\varphi_{12}\varphi_{13}\varphi_{15}=0, \tag{17} \]

will hold, from which the same conclusion will follow for the binary scale of the variables \((t_1,\tau_1)\); an analogous result will hold for each of the other pairs of variables.

The application of Lemma 1 leads to the following theorem.

Theorem 4. If a nonalgebraizable equation (1) admits nonprojective anamorphoses (2), then each of the binary fields degenerates into a binary scale on an elliptic curve of the third order, common to all three pairs of variables.

Consequently, the following theorem is valid:

General theorem. If an analytic nondegenerate equation (1) admits nonprojective anamorphoses of the form (2), then all the binary fields \((t_i,\tau_i)\) degenerate into binary scales on algebraic curves; the sum of the orders of the distinct curves in each anamorphosis is equal to three. If such an equation is algebraizable with respect to at least one of the variables, then it is similar to an equation of the third \(N\)-order and, consequently, admits anamorphoses of all three genera; in its nomograms of the third genus with a common base there serves a universal curve of the third order. If, however, equation (1) is not algebraizable, then it admits only nomograms of the third genus with a common base on an elliptic curve of the third order.

Remark. Using the parametrization of an elliptic curve by Weierstrass functions and Gronwall’s work \({}^{2}\), we conclude that equation (1) admits nonprojective anamorphoses if and only if it is either similar to an equation of the third \(N\)-order or similar to an equation equivalent to it.

Ural Polytechnic Institute
named after S. M. Kirov

Received
6 XI 1956

References

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