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Problem Set 1 The next several problems are concerned with Sylvester's matrix and
 

Summary: Problem Set 1
The next several problems are concerned with Sylvester's matrix and
the Sylvester determinant. Assume coecients are drawn from some xed
algebraically closed eld.
Problem 1. Let F1 = anxn
+ an-1xn-1
+ + a1x + a0 and F2 = bmxm
+
bm-1xm-1
+ + b0. Show that F1 and F2 share a root if and only if there
exists G, H with deg(G) = m - 1, deg(H) = n - 1 and GF1 = HF2.
Problem 2. Let S = {F1, xF1, . . . , xm-1
F1, F2, xF2, . . . , xn-1
F2} and let Pt
denote the vector space of all polynomials of degree less than or equal to t.
Use the result in Problem 1 to show that S is a linearly dependent set of
vectors in Pm+n-1 if and only if F1 and F2 share a root.
Problem 3. By writing each element of S in terms of the basis {1, x, x2
, . . . , xm+n-1
},

  

Source: Abo, Hirotachi - Department of Mathematics, University of Idaho

 

Collections: Mathematics