From: "David L. Johnson" Subject: Re: 4th order polynomial Date: Tue, 18 Jan 2000 21:11:42 -0500 Newsgroups: sci.math.symbolic Summary: [missing] "Eric J. Clavelle" wrote: > > Hello, > For a 4th order polynomial ax^4+bx^3+cx^2+dx+e=0 is there a way to > determine using the coefficients how many (if any) real roots there will > be? Of course, the next question is whether there is a nice way of > determining what these roots are (not numerically). Of courese, there is a 4th degree formula, determining solutions in terms of algebraic expressions in a...e. It is not pretty, but it certainly exists. > I suppose what i am looking for is the equivalent test for a 4th > order polynomial as there is for a cubic, ie. if ((2a-9ab+27c)/54)^2 < > ((a^2-3b)/9)^3 then the cubic x^3+ax^2+bx+c has 3 real roots (this being > from Numerical Recipes)? Well, knowing the expressions for the roots should give similar inequalities, since those are simply reality conditions for various of the roots involved in the cubic formula. Look at http://www.hsu.edu/faculty/worthf/cubic.html -- David L. Johnson david.johnson@lehigh.edu Department of Mathematics http://www.lehigh.edu/~dlj0/dlj0.html Lehigh University 14 E. Packer Avenue (610) 758-3759 Bethlehem, PA 18015-3174 Some people used to claim that, if enough monkeys sat in front of enough typewriters and typed long enough, eventually one of them would reproduce the collected works of Shakespeare. The internet has proven this not to be the case. ============================================================================== From: lrudolph@panix.com (Lee Rudolph) Subject: Re: Cubic Formula? Date: 6 Jul 2000 09:47:18 -0400 Newsgroups: sci.math Richard Carr writes: >What's a depressed cubic? In general, a monic n-ic in which the coefficient of the (n-1)st power of the variable is 0 is called "depressed". Thus a depressed monic cubic is x^3+px+q. Of course in characteristic 0 any monic n-ic can be transformed to a depressed monic n-ic by use of the appropriate Tschirnhausen transformation (i.e., translation of the variable). On the level of roots, a depressed monic n-ic is one whose roots sum to 0. The Tschirnhausen trick shows that the "depressed configuration space" of those unordered n-tuples of pairwise distinct complex numbers which sum to 0 is a deformation retract of the (usual, boomps-a-daisy, not-a-care-in-the-world) configuration space of all unordered n-tuples of pairwise distinct complex numbers, which is the usual model of the Eilenberg-MacLane space K(B_n,1) for the n-string braid group. Since the depressed configuration space has lower dimension (by two, counting real dimensions), this trick can sometimes be of minor assistance. For instance, it lets you see (without having to look at group presentations and know about asphericity of knots and all that) that the complement of a trefoil knot in the 3-sphere is a K(B_3,1). (Well, you still have to do *some* work.) Lee Rudolph ============================================================================== oops ============================================================================== From: lrudolph@panix.com (Lee Rudolph) Subject: Re: Cubic Formula? Date: 6 Jul 2000 12:20:26 -0400 Newsgroups: sci.math Richard Carr writes: >On 6 Jul 2000, Lee Rudolph wrote: ... >:On the level of roots, a depressed monic n-ic is one whose roots >:sum to 0. Since (x-r_1)(x-r_2)...(x-r_n) = x^n-(r_1+...+r_n)x^{n-1}+... . >:The Tschirnhausen trick shows that the "depressed >:configuration space" of those unordered n-tuples of pairwise >:distinct complex numbers which sum to 0 is a deformation retract >:of the (usual, boomps-a-daisy, not-a-care-in-the-world) configuration >:space of all unordered n-tuples of pairwise distinct complex numbers, More explicitly: Let r = (r_1+...+r_n)/n. Let r_k(t) = r_k-tr. Then t |-> {r_1(t),...,r_n(t)} parametrizes a path from [0,1] into the configuration space of all unordered n-tuples of pairwise distinct complex numbers (provided that {r_1,...,r_n} is in that space) which starts at {r_1,...,r_n} and ends at a point of the depressed configuration space. The depressed monic polynomial with roots {r_1(1),...,r_n(1)} is the Tschirnhausen transform of the monic polynomial with roots {r_1,...,r_n}. The paths give the advertised deformation retraction. >:which is the usual model of the Eilenberg-MacLane space K(B_n,1) >:for the n-string braid group. That is, the fundamental group of the configuration space is (by *definition* of B_n; at least, by the best definition...) B_n, and all higher homotopy groups of the configuration space are 0. This latter fact is a theorem of Faddell and Neuwirth. It is not hard to prove either with or without the machinery they used. >:Since the depressed configuration >:space has lower dimension (by two, counting real dimensions), this >:trick can sometimes be of minor assistance. For instance, it >:lets you see (without having to look at group presentations >:and know about asphericity of knots and all that) that the >:complement of a trefoil knot in the 3-sphere is a K(B_3,1). >:(Well, you still have to do *some* work.) As is well-known, the discriminant of the depressed cubic x^3+px+q (coefficients in C, say) is z^2-w^3, where z and w are suitable scalar multiples of p and q respectively. Thus the depressed configuration space (of three points) is the complement in C^2 (with linear coordinates z and w) of the cuspidal cubic z^2=w^3. This curve is the real cone (in non-linear real coordinates) on its intersection with the 3-sphere |z|^2+|w|^2=1, which is immediately seen to be a trefoil knot; so the depressed configuration space is the product of R_+ and the complement of the trefoil in the 3-sphere. >You've totally lost me here- actually fairly early on so I haven't read >the rest (but I assume you've totally lost me). Found, or still foundering? Lee Rudolph