Solutions of system of nonlinear equations

[smallphi]

Suppose you have N unknowns and N NON-linear equations of those unknowns. Is it possible that the equations are LINEARLY-independent, yet you get an infinite number of solutions?

I know the question of how many solutions you would get for a system of LINEAR equations is resolved with ranks of matrices. Is there an analogous treatment for system of NON-linear equations to get the number of solutions without actually solving the system?

References please.

 

[HallsofIvy]

What do you mean by "linearly independent" for non-linear equations?

 

[smallphi]

Linearly independent means that a linear combination of the equations doesn't give the equation 0 = 0 unless all the coefficients of the combination are zero. That is equivalent to saying that none of the equations is a linear combination of the others.

 

[smallphi]

My question is generated from the usual sloppiness in physics in counting degrees of freedom or in sloppy math textbook counting free parameters. You have N variables and M constraint equations for those variables. The usual conclusion is that if the constraints are 'independent', you are left with N-M free variables (degrees of freedom).

I can see how that goes for linear constraint equations but I'm not convinced it is the same case for nonlinear.

 

[DeadWolfe]

Well, the real problem here is what is meant by degrees of freedom. What you really want is for the solution set to for an n-m dimensional manifold, and one proves this works via the inverse function theorem.

 

[mathwonk]

this arises in algebraic geometry, and is called the phenomenon of excess intersection.

it is prudent to look at homogeneous equations in projective space, to best mimic the phenomenon of linear equations always having a trivial solution, since homogrneous equations always vanish at the origin.

so the question is to determine the number of intersections of n hyopersurfaces, i.e. the number of common solutions to n homogeneous equations, in projective n space, say over the complex numbers, so as to guarantee that each one of the equations has solutions.

the first theorem says that n hypersurfaces in P^n do have at eklast one intersection, analogous to the theorem that n linear equations in n+1 variables have a non trivial solution. but how many?

then bezouts theorem says one should expect the number of solutions to be the product of the degrees of the equations, but it can happen that the number is actually infinite.

a famous example concerns the problem of determining how many conics are tangent to 5 given ones in the plane. the space of plane conics is itself a projctive space of dimension 5, since there are 6 independent quadratic monomials in three variables.

the condition to be tangent to a given conic is given by a single equation of degree 6, hence each given conic defiens a hypersurface of degree 6 in P^5.

so their intersection shoud be those conics tangent to all 5 given ones, which should give 6^5 such conics. But notice that a double line is a conic which is tangent to every conic. and there is a whole surface of doubled lines, one for each line in the plane.

so we have 5 hypersurfaces in P^5 that always contain a common surface. we want to throw out this surface of phony solutions and count instead how many actual geometric solutions there are, the number of reduced conics that meet all 5 given ones.

it turns out in general there are 3264 I believe, except in characteristic 2 where there are 51.

this is a fascinating branch of algebraic geometry.

when n hypersurfaces in P^n meet only a finite number of times, their intersection is called "proper" and then bezouts theorem does hold, if the intersections are "counted correctly" in the same sense that one counts multiple roots of polynomials multiply in order to make the fundamental theorem of algebra true.

i.e. there is a theory of multiplicity of isolated solutions, such that the sum of the multiplicities of all solutions equals the product of the degrees of the hypersurfaces, IF there are only a finite number of such solutions.

One might ask what can one say about the situation when there are infinitely many? and one may ask as you did, when are there only finitely many?

 

[mathwonk]

well this somewhat frustrating browser just trashed my second post on the topic, but let me just refer you to the book, Intersection Theory, by William Fulton, and to his more elementary monograph on the subject from a conference at George Mason University about 25 yers ago. the phenomenon you ask about is called "excess intersection".

if you know about the fibers of a mapping, imagine a surjective mapping between spaces of the same dimension, in which most fibers are finite (and of the same cardinality = degree of the map) but some are infinite, and ask how to tell the difference.

you can already see geometrically that the images of these infinite fibers will have codimension 2 or more, whereas the points with finite fibers which have too
few points will have codimension one, (expected).

my thesis concerned the problem of computing the degree of a map from looking at infinitesimal neighborhoods of infinite fibers.

 

[mathwonk]

deadwolfe is discussing the theorem about dimension of images of TRANSVERSE maps. not all maps are tranverse everywhere.

transversality generalizes the notion of "regular" values, which is more commonly used.

 

[mathwonk]

here is a cute example due to beniamino segre. it is known that a general cubic surface in projective 3 space contains 27 lines. i.e. the degree of the map from the space of pairs (S,L) where S is a cubic surface and L is a line on S, projecting noto the surface S in the space of cubics, has degree 27.

but how to see this? consider the cubic surface A consisting of three planes, which of cousre conytains infinitely many lines. but choose another general cubic surface S and look at the line of surfaces defined by A+tS, and let t go to zero.

then we have a general surface A+tS with a finite number of lines approaching the surface A. Hence if we can compute which lines on A are limits of lines from A+tS, we should find the generic number.

Severi argues that these are exactly the lines passing through pairs of points where the cubic S meets the three coordinate lines of the three planes making up A. Since S meets each coordinate line 3 times, to get a limiting line we choose one point on one coordinate line and one on another, hence there are 27 of them!

 

[mathwonk]

here is another similar incantation: consider a cubic surface A formed of a quadric surface and a plane, which meet in a plane conic curve.

As t goes to zero, now the limiting surface A+tS has lines which approach those lines on A which join two points of Smeetplanemeet conic, or which lie on the quadric and pass through a point of Smeetplanemeet conic.

Since Smeetplanemeet conic is 6 points, there are 15 lines in the plane and 12 on the quadric, for a total of 27 again.

(A quadric surface is doubly ruled, with each point lying on two lines, so 6 points determine 12 lines.)

 

[mathwonk]

how do you like this, small phi?

 

[mathwonk]

the general theorem for dimensions, i.e. number of parameters is this: if you have a proper surjective map from an algebraic variety of dimension m to one of dimension n, e.g. any regular dominant map of projective algebraic varieties, then m is at least n, and the dimension of every fiber is at least m-n.

moreover, for any k, the set of points of the target where the fiber has dimension at least k, is closed.

furthermore, if k >0, the set where the fiber has dimension at least m-n+k has codimension at least k+1 in the target.

 

[mathwonk]

ok here is deadwolfe's inverse function theorem version: given n hypersurfaces in projective complex n space, assume that at every intersection point, their gradients are linearly independent, i.e,. they are all non singular manifoklds at each intersection point, and at each one their tangent spaces intersect in only one point.

then the intersection points are called transversal, and each counts with multiplicity one. They are all isolated and there is a finite number of them, equal precisely to the product of the degrees of their defining homogeneous polynomials.the mor interesting cases however are the finite but non transverse intersections.

for example suppose a plane quartic curve has three non collinear double points. then the space of plane conics that all pass through all three of them form a projective plane, and give a map of the original plane to a new plane ,such that the image of the given curve is a conic.

why? because the conics in tersect the quartic with total algebraic multiplicity 8, but 6 are accounted for by the thre double points. this leaves two further intersection points.

over in the target plane, the conics beome coordinate functions, hence transform to lines, and so the two further intersections mean that the quartic has become a conic.

this proves that a quartic with thre general double points is almost isomorphic to a conic, in particular has geometric genus zero, whereas a smooth quartic has genus three.

not surprisingly those three double points each swallowed one handle from the genus three riemann surface.

i love this stuff. elementary references include the clasic book algebraic curves by walker, the modern one by fulton, and the recent one by gerd fischer. also a big tome by brieskorn and knorrer, which is just marvellous, written in the old style with ample pictures and lots of history, and coming up to the modern ideas as well, in about 800 pages of loving discussion.

 

[mathwonk]

the genberal question asked here, how many intersections do several non oinear equations have? is almost the general subject of algebraic geometry: i.e. describe the common solutions of systems of non linear algebraic equations. this subject is slightly analogous to linear algebra but so much harder and more interesting.

as seen here, one technique is to reduce to linear algebra as in calculus by applying linear algebra hypotheses to the approximating tangent hyperplanes of the hypersurfaces, and then see what comes out. but the phenomenon of non transverse intersections is new and holds the most interesting part of the story.

i.e. there are no singularities in inear algebra, except for perhaps the consideration of multiple roots of characteristic polynomials, which also yields the most interesting part of that story, namely non diagonalizable matrices.

 

[mathwonk]

here is a nice simple example, just take a saddle surface xy-z^2 =0, a smooth quadric of rank 4, and intersect it with a tangent plane z=0. you get xy=0, which is 2 lines in the plane z=0.

so the intersection is reducible of degree 2. in fact every tangent plane to this quadric cuts two lines. why? well it cuts a quadric curve but one with a singular point at the point of tangency, and a singular conic is two lines! thus every smooth quadric aurface is doubly ruled. and in fact isomorphic to P^1 x P^1.