From: rusin@vesuvius.math.niu.edu (Dave Rusin) Newsgroups: sci.math Subject: Re: Complex Vars and Solving Systems of Higher Ordered Eqns Date: 3 Dec 1998 17:54:09 GMT Fran Brenneman wrote: >I have a question as to whether anyone has ever looked at the use of >complex variables to solve systems of higher ordered eqns. [Example given expresses the single complex equation w^6=z as a pair of real equations in four real variables.] >So this leads to the question of whether it is possible to analyze or >solve a system of higher ordered polynomials in 2 variables using >complex analysis in this way (i.e. whether it is possible to recast a >system of 2 higher ordered polynomial eqns which may be impossible to >solve by algebraic means in terms of the nth root of some complex number >which we can do much more easily)? The answer is: sometimes yes, sometimes no. When it works, it seems fairly clear that this process would be easier; the analysis of a system of N equations in M unknowns should be easier than that of 2N equations in 2M unknowns, even if those unknowns and equations are complex instead of real (say). Indeed, in a lot of problems it becomes easier to look for complex solutions anyway, and throw away any that aren't real, if you should only be interested in the real ones in the first place. Unfortunately, not all sets of equations permit this kind of reduction. Your equations will describe an (analytic or algebraic) variety of even dimension in R^(2M), and you want to know whether this is may be given a complex structure. But the underlying real varieties of complex varieties are a very special class of real varieties. For example, nonsingular complex varieties are always _orientable_ manifolds, so a set of equations whose solution set is the Klein bottle cannot be contracted to a set of complex equations in half as many variables. The situation can be characterized a bit more algebraically if you ask whether the equations themselves, rather than their solution set, come directly from reading off real and imaginary parts of complex equations. If f(z1, z2, ...) = 0 is an equation relating complex variables, and f is complex-analytic (e.g. a polynomial), then the real and imaginary parts of this equation are bound by the Cauchy-Riemann equations: if z_n = x_n + i y_n and f = u + i v, then we must have equality of all the partial derivatives: d u / d x_n = d v / d y_n and d v / d x_n = - d u / d y_n Note in particular that this means u (as well as v) is _harmonic_ : it satisfies the differential equation d^2 u / d^2 x_n + d^2 u / d^2 y_n = 0, which means some sets of real equations u = 0 cannot even arise from taking _linear combinations of_ the various real and imaginary parts. This material arises at the beginning of a Complex Analysis course: index/30-XX.html Discussion of analytic varieties would be part of Several Complex Variables: index/32-XX.html (What the heck: let's mention the small field of Potential Theory, index/31-XX.html which amounts to the study of harmonic equations.) dave ============================================================================== From: wcw@math.psu.edu (William C Waterhouse) Newsgroups: sci.math Subject: Re: Complex Vars and Solving Systems of Higher Ordered Eqns Date: 3 Dec 1998 23:19:18 GMT In article <366399DE.C70B1663@netscape.net>, writes: > I have a question as to whether anyone has ever looked at the use of > complex variables to solve systems of higher ordered eqns. > > I have in mind a very specific application of this idea. > Consider the soln to the eqn w^n = z, where z is a known complex number > and w is therefore (in general) one of the n nth roots of z. Given z and > expressing z in polar form, the solution for w is in general > straightforward. But if we express w in the form: > (i) w = x + i y > we can also attempt to solve for w by solving for x and y via the > following 2 eqns : > (ii.a) Re[ w^n ] = Re[z] > (ii.b) Im[ w^n ] = Im[z] > which are entirely in terms of the real variables x and y. >... > So this leads to the question of whether it is possible to analyze or > solve a system of higher ordered polynomials in 2 variables using > complex analysis in this way (i.e. whether it is possible to recast a > system of 2 higher ordered polynomial eqns which may be impossible to > solve by algebraic means in terms of the nth root of some complex number > which we can do much more easily)? >... > Matthew T. Brenneman This seems not to have received any answer yet, so I'll say a bit. There is nothing wrong with the basic idea, but the problem is that the pairs of real equations that arise in this way are so special that they would almost never occur on their own. To see what's happening, let's consider any complex polynomial f(z) = f(x+iy) = P(x,y) + iQ(x,y) where the P and Q are real polynomials. Suppose first f has just one term, f = (a+ib)(x+iy)^k. Then f_x = the partial derivative of f with respect to x is of course k(a+ib)(x+iy)^{k-1}, by the chain rule. Similarly, the other partial derivative f_y equals ik(a+ib)(x+iy)^{k-1}. Thus we get f_y = i f_x. This will be true for each term, so it is true for any polynomial f(x+iy). Writing this out in terms of P and Q, we have i(P_x + i Q_x) = P_y + iQ_y. Taking real and imaginary parts, we get relations on the partial derivatives (called the "Cauchy-Riemann equations"): P_x = Q_y, P_y = - Q_x. Differentiating again, we get the second derivatives satisfying f_{yy} = i^2 f_{xx} = -f_{xx}, so we get P_{yy} = - P_{xx} and Q_{yy} = -Q_{xx}. These equations tell us that the polynomial P(x,y) must satisfy the differential equation P_{xx} + P_{yy} = 0, which is a strong requirement. And once we have some P satisfying this condition, then we can determine both Q_x and Q_y from the corresponding partial derivatives of P, so that P determines Q completely except for its constant term. It is true, in fact, that these are the precise requirements: that is, if you have a real P(x,y) satisfying P_{xx} + P_{yy} = 0, then there is a complex polynomial f(z) (unique up to constant) with real part equal to P. William C. Waterhouse Penn State