From: wilker@math.purdue.edu (Clarence Wilkerson) Subject: Re: Linear factor of a multivariate polynomial Date: 30 Jan 1999 06:46:51 GMT Newsgroups: sci.math.symbolic,sci.math.research Keywords: When does a polynomial split completely into linear factors? >> Gudbjorn Freyr Jonsson asks: >>I'd like to know when a polynomial in more than one variable has a >>linear factor. Here's a solution to a different problem: for factoring over the rationals or integers a homogeneous polynomial in several variables completely into linear factors: Let F be the form in Q[x_1,x_2,..x_n]. For all large primes p you must check that in F_p[x_1,x_2,..x_n] one has F | P^i F for all i > 0 (actually only have to check up to i < degree F ) Here P^i F is the i-th mod p Steenrod operation given by identifying F_p[x_1,x_2,..x_n] as the mod p cohomology of (CP^\infty})^n. More explicitly, P^i is determined by P^1x_j = x_j^p, P^0x_j = x_j, P^ix_j = 0 for i > 1 for all j. Extend to all of F_p[x_1,x_2,..x_n] by the "Liebnitz" or Cartan formula that P^k(uv) = \sum_{i=0}^k P^{i}uP^{k-i}v This appears in Wilkerson, Clarence Classifying spaces, Steenrod operations and algebraic closure. Topology 16 (1977), no. 3, 227--237. I don't know if this corresponds to any classical technique or not. One can simplify the checking a bit by using the total Steenrod operation P_t f= \sum_0^infty P^k(f)t^k Then P_t(fg) = P_t(f)P_t(g) Then one just has to check that F | P_t(F) in F_p[x_1,x_2,..x_n][t]. But P_t(F) is just F( (x_1 + tx_1^p), ... (x_n + tx_n^p)]. This is related to work of Serre: Serre, Jean-Pierre Sur la dimension cohomologique des groupes profinis. (French) Topology 3 1965 413--420. -- Clarence Wilkerson \ HomePage: http://www.math.purdue.edu/~wilker Prof. of Math. \ Internet: wilker@NOspam.math.purdue.edu Dept. of Mathematics \ Messages: (765) 494-1903, FAX 494-0548 Purdue University, \ W. Lafayette, IN 47907-1395 \ ============================================================================== From: rusin@math.niu.edu (Dave Rusin) Subject: Re: Linear factor of a multivariate polynomial Date: 2 Feb 1999 07:47:09 GMT Newsgroups: sci.math.symbolic,sci.math.research Gudbjorn Freyr Jonsson asks: >I'd like to know when a polynomial in more than one variable has a >linear factor. Clarence Wilkerson wrote: >Here's a solution to a different problem: > >for factoring over the rationals or integers >a homogeneous polynomial in several variables completely >into linear factors: I'm not sure if Clarence intended only to look for polynomials with _all_ factors linear, but the same idea tests for the presence of _any_ linear factors. If L is a linear factor of F, then L must divide P(F) so we can compute putative linear factors L (modulo any prime with good reduction) as divisors of gcd(F, P^i(F)). For example, when p=2 and F = x^2 + y^2 + xy we compute P(F) = (x+x^2)^2 + (y+y^2)^2 + (x+x^2)(y+y^2) = (x^2+xy+y^2) + (x^2+xy^2) + (x^2+xy+y^2)^2 but as gcd( x^2+y^2+xy, xy(x+y) ) = 1, we conclude x^2+xy+y^2 has no linear factors. But fairly efficient polynomial factorization procedures exist, as can be seen by asking your favorite symbolic-algebra program to factor a polynomial. I don't know if it's any easier, in general, to restrict attention to linear factors, although the Steenrod-algebra approach above is limited to this case. Check out books on algorithmic aspects of algebra and number theory, e.g. at http://www.math.niu.edu/~rusin/known-math/index/12FXX.html dave