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Software tools for visualizing multivalued functions.

The Challenge of Multi-valued Arithmetics

Perhaps, in the first time students meet with the concept of multivalued functions when dealing with the quadratic formula. The second encounter with the idea occurs in high school, while studying the notion of roots of complex numbers. This subject becomes especially important in the algebra of polynomial equations and in the complex analysis.

Life in the world of multi-valued functions is hard: even adding them is a questionable operation! Just think how would you interpret, say, the formula [square root of 2 + 3i] + [square root of 1 - i]? Does it represent four complex numbers--all the possible sums of a number from the two-element set [square root of 2 + 3i] and a number from the two-element set [square root of 1 - i]? If so, how about the product [square root of 2 + 3i] * [square root of 1 - i] That set must consist only of two elements because ([+ or -]1)*([+ or -]1) = ([+ or -]1)*([+ or -]1) = ([+ or -]1)! Along the same lines, is it true that [square root of z] + [square root of z] = 2[square root of z] ? Well, if [square root of z] takes two values, say a and -a, then the possibilities for [square root of [square root of z] + [square root of z] are: 2a, -2a, and 0 = a + (-a). On the other hand, the possibilities for 2yfz are only 2a and -2a.

So the standard algebraic rules for operating with regular functions fail miserably when dealing with multi-valued ones.

Domain-Range Visual Representations of Multi-valued Functions

In order to visualize a map f C [right arrow] C of complex planes on a computer screen, we separate the domain and range of f and show them in two different views-windows [9]. This presentation helps to visualize the correspondence between the source z and the target(s) fz). For example, we can use the colour coding to indicate the correspondence (compare this interface with [4], [10] and [12], where only the range window is available).

Let us start with the example of how VisuMatica represents the multi-valued function fz) = [fifth root of z] (it describes all the solutions of the complex equation [w.sup.5] = z for a variety of complex numbers z) (see Fig. 1). As we move the cursor over the point z of the complex plane (the domain window), the set of values [fifth root of z] is displayed in a different complex plane (the range window). The color-coding emphasizes the five-fold ambiguity in determining the value of f in the range. Note that the five images of z share their colour with z (currently yellow).

Occasionally, a dynamic visualization of multiple values of a function becomes difficult if done by moving the mouse manually. In such cases, we can specify the mouse trajectory analytically. For example, consider the 6-valued function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] The task is to see the trajectories traced by its multiple

values, as the pointer at z moves, say along the circle of radius 1.25, centred on the origin. Fig. 3 shows the image trajectories.

Using the "Simulation" tool of the "Mapping" dialog box and moving the slider (marked by an arrow in Fig. 3), the user can control the movement of z in the domain and of its image(s)f z) in the range. The value of the polar coordinate of z is automatically updated in the textbox on the right side of the slider. One can type in this textbox the value of an angle; the pointer and its corresponding six images (the small red circles) will appear in the Range window.

Multi-valued Functions and Riemann Surfaces

Problems with the arithmetic of multi-valued functions have an elegant solution. In the case of multi-valued analytic functions of a single complex variable, the proposed solution is called "The Riemann Surface'. On technical level, the construction of Riemann surfaces is based on the principle of analytic continuation. However, here we adopt a more geometrical viewpoint. In any case, our goal here is not to introduce the reader to the classical theory of Riemann surfaces (see [2], [8]), but to show the usefulness of VisuMatica in generating expressive dynamic images, the images that facilitate our understanding of the subject.

At the first glance, the idea of a Riemann surface looks as a cheap trick: we replace the domain of a given multi-valued function by a new more complex domain on which the function becomes single-valued.

Let us illustrate this trick with the simplest possible example. Take the 2-valued function f (z) = [square root of z] of the complex variable z. Consider the complex algebraic curve (the real surface) [summation] in [C.sup.2] whose equation is {[w.sup.2] = z}, a complex parabola. If w = u + iv and z = x + iy, then, in the real uvxy-space, [summation] can be also described by two quadratic polynomial equations: {x = [u.sup.2] - [v.sup.2], y = 2uv}.

As we try to express w in terms of z, we get w = [square root of z], our original 2-valued function.

Now replace C, the domain of [square root of z], with the curve/surface [summation], and f (z) = [square root of z] with the coordinate w, viewed as a single-valued function on [summation]. Note that the coordinate w is a map from [C.sup.2] to C and thus can be applied to any subset in its domain, including the complex curve [summation]. This complex curve [summation], equipped with the coordinate maps z: [summation] [right arrow] C and w: [summation] [right arrow] C, is the Riemann surface of the function [square root of z].

So how can we visualize this and other Riemann surfaces? Since we do not have the gift of seeing a 2-dimensional object in a 4D-space, we need to downgrade our ambitions to 2-dimensional objects in 3D. Here is where VisuMatica becomes handy. Of course, such a 3-dimensional picture will not give a faithful representation of the way [summation] resides in [C.sup.2]. Nevertheless, it will be very useful for understanding the geometry of [summation] in connection to both coordinates z and w in [C.sup.2]. Moreover, an appropriate pair of such 3D-images will constitute a faithful representation of [summation] in [C.sup.2].

Let us examine a more general setting. Let G(z, w) be a complex polynomial of degree d in the variables z and w. Consider an algebraic curve (a real surface) [summation] whose equation in [C.sup.2] is G(z, w) = 0. How is it possible to draw [summation]?

Here is the main idea behind traditional depictions of [summation] in 3D. Let [F.sub.a] stand for the intersection of [summation] with the complex line {z = a}. If a polynomial G(z, w) of degree d is not divisible by (z - a), then [F.sub.a] is a finite set of cardinality d at most. Consider w([F.sub.a]), the w-image of [F.sub.a] in C. Pick one of the following standard coordinate functions x: C [right arrow] R or y: C [right arrow] R (as usually, x(z) = Re(z) andy(z) = Im(z)). For each z in C, apply the function x (or y) to the set w([F.sub.z]) in C to produce the set x(w([F.sub.z])) in R. Let V be the real 3D-space of pairs (z, t), where z is a complex number and t is a real number. Consider a surface Zx in V, formed by the pairs (z, t), where z runs over C and t belongs to the finite set x(w([F.sub.z])) "suspended over z". Similarly, employing the function y: C [right arrow] R, we can construct another surface [[summation].sub.y] in V. We view both surfaces, [[SIGMA].sub.x] and [[summation].sub.y], as representing the same Riemann surface Z. Each of them gives a lot of information about Z; together they form its faithful representation.

Riemann surfaces Zhave an interesting topology. It can be recovered from their images (like Z) in 3D. For a generic degree dpolynomial G(z,w), topologically [summation] is a 2-sphere to which (d - 1)(d - 2)/2 handles have been attached and from which d points have been deleted (see the left picture in Fig. 4). In particular, for d = 2, we get a 2-sphere with two punctures; for d = 3, we get a torus (the surface of a bagel) with three punctures. When [summation] has singularities (for example, simple self-intersections), its topology is more complex (see the right picture in Fig. 4). In such a case, [summation] can have singularities that are obtained by identifying a number of points on a punctured 2-sphere with (d - 1)(d - 2)/2 handles into a single point. For example, in the vicinity of a simple self-intersection point, [summation] looks like a double cone (consider the vicinity of bold black points in Fig.4).

Fig. 5 shows the surface [[SIGMA].sub.x] for the familiar quadratic parabola [summation] = {z - w = 0}, the Riemann surface of the multi-valued function [z.sup.{1/2}]. Even in this most basic case, reconstructing the topological type of [summation] from these pictures takes some thought. In fact, [summation] is a 2-disk, or a 2-sphere with a single puncture.

Often it is more expressive and informative to show the intersections of the surfaces [[SIGMA].sub.x] or [[summation].sub.y] with the surface of a square prism {[parallel]Re(z)[parallel] [parallel] [less than or equal to] r, [parallel]Im(z)[parallel] [less than or equal to] r} (see Fig. 5, (a) and (b)) or with the surface of a circular cylinder {[parallel]z[parallel] [less than or equal to] r} (see Fig. 5 (c) and (d)), both residing in the 3D-space V = [C.sup.1.sub.z] x [[R.sub.1.sub.t].

In Fig. 5, (b), the intersections of [[SIGMA].sub.x] with the surface of [begin strikethrough]the[[end strikethrough] prism is shown as a narrow self-intersecting colourful band, while the intersections of [[SIGMA].sub.x] with the prism itself look semitransparent.

In the case of [begin strikethrough]a[end strikethrough] circular cylinder, we show (diagram (c)) only the intersection Q of [[SIGMA].sub.x] with the surface H of the cylinder. Cutting H along a vertical line opens H into a rectangle. The rectangle retains the intersection pattern Q. This pattern is especially informative when we are trying to count the number of loops in the boundary of the portion of [SIGMA] that is contained in the subset {(z, w): [parallel]z[parallel] [less than or equal to] r} of the space []. As you see, the intersection of [[SIGMA].sub.x] with the surface of the cylinder {(z, t): [parallel]z[parallel] [less than or equal to] r} forms a single self- intersecting loop.

It is clear from Fig. 5 that the behaviour of the projection [phi]: [[SIGMA].sub.x] [right arrow] [C.sup.1.sub.z] over the origin 0 is qualitatively different from its behaviour over any other point z in [C.sup.1.sub.z]. Such special point (like the origin) in the target space is called a ramification point of the map [phi]. The subset [F.sub.z] = [n.sub.-1](z) of [summation] is called the fiber of the map [phi] over the point z.

In fact, Riemann surfaces do not solve all our problems with multi-valued formulas in a single complex variable: for example, there is no simple connection between the Riemann surfaces of the multi-valued functions f(z) = [cube root of z] , g(z) = [square root of 1+z] and their sum h(z) = [cube root of z] + [square root of z]. The first one, [[SIGMA].sup.f] is the cubic parabola {[w.sup.3] = z}; the second one, [[SIGMA].sup.g], is the quadratic parabola {[w.sup.2] = z + 1}; the third one, [[SIGMA].sup.h], is an algebraic curve of degree 6 whose equation {[([w.sup.3] + 3zw + 3w - z).sup.2] = [(1 + z)(3[w.sup.2] + z + 1).sup.2]}we did not bother to simplify.

The three surfaces [[SIGMA].sup.f], [[SIGMA].sup.g], and [[SIGMA].sup.h], are not related in any obvious way (see Fig. 6 which shows the surfaces [[SIGMA].sup.g.sub.x], [[SIGMA].sup.g.sub.x], and [[SIGMA].sup.x.sub.x]), except for the evident relation between their ramification points. However, they testify that, for each z in [C.sup.1.sub.z], there is a peculiar way of continuous coupling of elements from the fiber [[SIGMA].sup.z.sub.f] (of cardinality [less than or equal to] 3) with the elements of the [[SIGMA].sup.z.sub.g] (of cardinality [less than or equal to] 2), so that the coupling produces elements of the fiber [[SIGMA].sup.z.sub.h] (of cardinality [less than or equal to] 6)! Note that there are exactly 6 handshakes between the members of a group of two friends and the members of a group of three friends...

Our hope is that creating graphical representations like Fig. 5 and Fig. 6 will help students to develop intuitive mental models of the ramification phenomenon, pivotal for the theory of multivalued complex functions.

Investigating the Quadratic Formula with VisuMatica

After getting somewhat familiar with the notion of a Riemann surface and the geometry of multi-valued functions it reflects, let us revisit the familiar quadratic formula - p/2 + [square root [p.sup.2]/4 - q]. Each of its two values satisfies the equation [w.sup.2] + pw + q = 0. We can study the surface S whose equation in the pqw-space (real or complex) is {[w.sup.2] + pw + q = 0} (see Fig. 7). The surface S is equipped with the projection n S [right arrow] [R.sup.2.sub.pq] (or n S [right arrow] [C.sup.2].sub.pq]), where [phi](p, q, w) = (p, q), and with the coordinate function w: S [right arrow] R (or w: S [right arrow] C). The construction of S resembles a high-dimensional analogue of the Riemann surface construction: one replaces the multi-valued formula f(p, q) = -p/2 + [square root of [P.sup.2]/4 - q] with the "honest" function w: S [right arrow] C.

Our graphics ability to show the projection of S onto [C.sup.2.sub.pq] is even more limited than our ability to portray Riemann surfaces. Nevertheless, since VisuMatica does a decent job of showing Riemann surfaces in 3D, we take advantage of this ability to get insights into the geometry of projection [phi]: S [right arrow] [C.sup.2.sub.pq].

Let us pick a complex line or a curve C in the pq-coefficient space. Consider the complex curve [summation] = [[phi].sup.-l] (C), the "slice" [summation] of S suspended over C. Then we have a chance to visualize the projection [phi]: [summation] [right arrow] C.

For example, take the line {p = const} for the role of C. In particular, the q-axis {p = 0} is a good example to start with. Then the quadratic formula collapses to f(0, q) = [square root of -q], and in the q-direction we see a copy of the familiar Riemann surface for the radical [square root of z]. The behaviour of n [phi]: [summation] [right arrow] C, where C is a line transversal to the discriminant curve D ={[p.sup.2]/ 4 q = 0} at a point (p, q), is similar.

For a generic line C, the map [phi]. [summation] [right arrow] C has two ramification points; these are the points where the line C hits the discriminant curve D. Fig. 7 shows the lift to [summation] of a loop [gamma] in C that captures: a) none of the ramification points (then [[phi].sup.-1]([sigma]) consists of two loops and [phi] is 1-to-1 on each of them), b) a single ramification point (then [[phi].sup.-1]([sigma])consists of a single loop and [phi] is 2-to-1 on it), c) both ramification points (again, [[phi].sup.-1]([sigma])) consists of two loops and [phi] is 1-to-1 on each of them). In Fig. 7, we have chosen the line {q = 1+ i} for the role of C.

In fact, outside VisuMatica, we do not know about any attempts to visualize the geometry of complex quadratic formula.

Visualizing the Cardano Cubic Formula and its [S.sub.3]-symmetry

In a similar spirit, let us investigate the solutions of depressed cubic equations [z.sup.3] + pz + q = 0.

The substitution z = w - 3p/w transforms this equation into an auxiliary equation [w.sup.6] + q[w.sup.3] - [p.sup.3]/27 = 0 of degree six. The latter admits solutions in radicals. In fact, its general solution w is given by the formula [alpha](p, q) = [-q/2 + [[([q.sup.2]/4 + [p.sup.3]/27).sup.{1/2}]].sup.{1/3}], a 6-valued function in the complex variables p, q. Moreover, any solution z of the original cubic equation is given by the Cardano Formula [alpha](p, q) - 3p/ [alpha](p, q) (see [2]).

Let us explore the geometry and symmetry of the Cardano formula with VisuMatica.

Fig. 8 shows a remarkably symmetric pattern of values {[alpha](p, q)}, where q = Ap + B. Note that the six values of a split into two triples, each of which forms a regular triangle. The two triangles can be exchanged by a rational linear transformation of the complex plane, a transformation which resembles the inversion map with respect to a given circle. In fact, the action of [S.sub.3], the permutation group in tree elements, preserves the configuration of six points in Fig. 8.

It is interesting to observe that the three solutions of the original cubic equation [z.sup.3] + pz + q = 0 do not exhibit any evident symmetry. In a sense, it is the hidden [S.sup.3]-symmetry of the auxiliary equation that allows for the solution of the original cubic equation in radicals!

Let us take these observations further.

Denote by S the surface {[z.sup.3] + pz + q = 0} in and by W the surface {[w.sup.6] + q[w.sup.3]-[p.sup.3]/27 = 0}.

Both surfaces project on the coefficient plane [C.sup.2.sub.pq]. Moreover, the substitution z = w 3p/w gives rise to a partially defined 2-to-1 ramified map k: W [right arrow] S

As in case with the quadratic formula, we will visualize complex slices S and W of S and Wthat are suspended over a typical line C in [C.sup.2.sub.pq]. Fig. 10 (cf. Fig. 3) shows the Riemann surface of the complex function [alpha](p, p - i). Fig. 11 depicts the nature of the map [pi]: W [right arrow] C in the vicinity of its ramification points. For generic line C, they are the three points where C hits the complex discriminant curve D = {[q.sup.2]/4 + [p.sup.3]/27 = 0} (see Fig. 9 for the real analogue of the curve D) and a singleton where C hits the coordinate line {p = 0}. These pictures help us to reconstruct the topology of a typical slice W, shown in Fig. 12 (see [6], [7] for more details). It exhibits a remarkable [S.sub.3]-symmetry, where [S.sub.3] denotes the permutation group in 3 symbols.

Fig. 11 and its variations help to analyse the topology of the slice W of the surface W = {[w.sup.6] + q[w.sup.3]-[p.sup.3]/27 = 0} over a generic line L in the coefficient plane [C.sup.2.sub.pq]. The next Fig. 12 depicts the ecstatically pleasing results of this analysis.

Of course, an algebraic geometer will easily determine this topology using some standard computational techniques ([2], [8]). Again, our point is that VisuMatica can replace these relatively advanced computations with dynamic interactive models that faithfully reflect them.

Note the complex curve W has two singularities one of which resides "at [infinity]". Each of the two singular points, 0 and [infinity], has a small neighbourhood in W whose the boundary consist of 3 loops (Fig. 11 shows these three loops in the vicinity of [infinity], and Fig. 12 delivers a "global view" of W and its singularities).

Visualizing the Monodromy of Riemann Surfaces while Exploring the Abel-Ruffini Theorem

In studying the solvability of polynomial equations P(z) = 0, it is important to understand the so called monodromies of the Riemann surface [[SIGMA].sub.P] associated with a given equation. As usual, we call VisuMatica for help.

In this context, monodromies are transformations of the fibers -[[phi].sup.-l] (p) of map [phi]. [[summation].sub.P] [right arrow] C, the transformations that occur as p traces a loop in C. The right diagrams in Figures 10 and 11 will give the reader an intuitive feel for the monodromy concept: just follow each curve on the surface of cylinders to see where it starts and ends on a fixed vertical line.

We will not attempt here to explain the Geometric Galois Theory (see [1], [6]) in its full generality. Instead, consider one specific p-family of equations: {[z.sup.5] - z + p = 0}.

The famous Abel-Ruffini Theorem claims that the equations {[z.sup.5] - z + p = 0} have no solutions in radicals for all, but finitely many values of p!

Let [SIGMA] be the curve {[z.sup.2] - z + p = 0} in [[C.sup.2].sub.pz] equipped with the projection n (z, p) -- p on the p-coordinate line C. This map [phi] [summation] [right arrow] C is ramified over four points [p.sub.1],[p.sub.2],[p.sub.3],[p.sub.4] in C. As we trace a loop [gamma] in the complement n to these four points, at each point p of [gamma], a permutation of the fiber [[phi].sup.-l](p) is generated (note that [[phi].sup.-l](p) is exactly the set of all five roots of the equation [z.sup.5] - z + p = 0). This permutation is an element of [S.sub.5], the permutation group in five elements (see Fig. 12).

An important step in proving the Abel-Ruffini Theorem is to check that any permutation from S5 can be realized as the monodromy of [phi]: [summation] [right arrow] along an appropriate loop [gamma] in n (an algebraist would say: "S5 is the Galois group of a generic equation [z.sup.5] - z + p = 0"). Eventually, this conclusion clashes with the assumption that there exists a formula in radicals that solves all equations of the form [z.sup.5] - z + p = 0.

The RHS diagram in Fig. 12 provides us with an effective visualization of the monodromy of the map [phi] [summation] [right arrow] C. This pattern of five strings is called a braid. The braids can be interpreted as elements of a group [B.sub.5], an extension of the permutation group [S.sub.5]. Braid groups play an important role in algebra, topology, and the singularity theory. So VisuMatica is able to portray braids well.

Concluding remarks

[6] and [1] contain a variety of projects and virtual experiments that extend the ones suggested by this paper, as well as an explanation of mathematical theory behind them. Although such theory resides on the border between undergraduate and graduate mathematics, the interactive use of VisuMatica makes it much more accessible to students with different mathematical backgrounds.


[1] Alekseev, V.B., Abel's Theorem in Problems and Solutions, Moscow State University, Moscow (1976).

[2] Brieskorn, E., Knorrer, H., Plane Algebraic Curves, Birkhauser Verlag, Basel-Boston-Stuttgart (1986).

[3] http://cinderella. de/tiki-index.php

[4] Farris, F., Visualizing complex-valued functions in the plane. MAA online,


[6] Katz, G., Nodelman, V., The Shape of Algebra in the Mirrors of Mathematics, World Scientific Publishing (2011).

[7] Katz, G., Nodelman, V., Computer-aided explorations of modular spaces of real polynomials: Giving geometric life to routine algebra The Electronic J. of Mathematics and Technology 5(1), 2011.

[8] Kirwan, F., Complex Algebraic Curves, London Mathematical Society, Cambridge University Press (1992).

[9] Nodelman, V., Multiple Views in Educational Software for Mathematics. Proc. The 2006 World Congress in Computer Science Computer Engineering, and Applied Computing - MSV'06. The 2006 Int'l. Conf.on Modeling, Simulation and Visualization Methods. Las Vegas, Nevada, USA, 26-29 June 2006 (210-213).

[10] Thaller, B., Visualization of Complex Functions. The Mathematica[R] Journal

[11] Wagon, S., Mathematica in Action. Problem Solving Through Visualization and Computation. 3ed, Springer (2010).

[12] Wegert, E., Visualization of Complex Functions. Plea for the Phase Plot (2009).

Gabriel Katz

Massachusetts Institute of Technology

Cambridge, MA 02139, USA

Vladimir Nodelman

Holon Institute of Technology

Holon, 52 Golomb St., 58102, Israel
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Author:Katz, Gabriel; Nodelman, Vladimir
Publication:Electronic Journal of Mathematics and Technology
Article Type:Report
Geographic Code:1USA
Date:Feb 1, 2013
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