FACTORING SPARSE POLYNOMIALS


 Amos Warren
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1 FACTORING SPARSE POLYNOMIALS
2 Theorem 1 (Schinzel): Let r be a positive integer, and fix nonzero integers a 0,..., a r. Let F (x 1,..., x r ) = a r x r + + a 1 x 1 + a 0. Then there exist finite sets S and T of matrices satisfying:
3 Theorem 1 (Schinzel): Let r be a positive integer, and fix nonzero integers a 0,..., a r. Let F (x 1,..., x r ) = a r x r + + a 1 x 1 + a 0. Then there exist finite sets S and T of matrices satisfying: (i) Each matrix in S or T is an r ρ matrix with integer entries and of rank ρ for some ρ r.
4 Theorem 1 (Schinzel): Let r be a positive integer, and fix nonzero integers a 0,..., a r. Let F (x 1,..., x r ) = a r x r + + a 1 x 1 + a 0. Then there exist finite sets S and T of matrices satisfying: (i) Each matrix in S or T is an r ρ matrix with integer entries and of rank ρ for some ρ r. (ii) The matrices in S and T are computable.
5 (iii) For every set of positive integers d 1,..., d r with d 1 < d 2 < < d r, the nonreciprocal part of F (x d 1,..., x d r) is reducible if and only if there is an r ρ matrix N in S and integers v 1,..., v ρ satisfying d 1 v 1 d 2. = N v 2. d r v ρ but there is no r ρ matrix M in T with ρ < ρ and no integers v 1,..., v ρ satisfying d v 1 1 d 2. = M v 2.. d r v ρ
6 the nonreciprocal part of F (x d 1,..., x d r) is reducible
7 the nonreciprocal part of F (x d 1,..., x d r) is reducible F (x 1,..., x r ) = a r x r + + a 1 x 1 + a 0
8 the nonreciprocal part of F (x d 1,..., x d r) is reducible F (x 1,..., x r ) = a r x r + + a 1 x 1 + a 0 F (x d 1,..., x d r ) = a r x d r + + a 1 x d 1 + a 0
9 Theorem 2 (Schinzel): Let r be a positive integer, and fix nonzero integers a 0,..., a r. Let F (x 1,..., x r ) = a r x r + + a 1 x 1 + a 0. Then there exist finite sets S and T of matrices satisfying:
10 Theorem 2 (Schinzel): Let r be a positive integer, and fix nonzero integers a 0,..., a r. Let F (x 1,..., x r ) = a r x r + + a 1 x 1 + a 0. Then there exist finite sets S and T of matrices satisfying: (i) Each matrix in S or T is an r ρ matrix with integer entries and of rank ρ for some ρ r.
11 Theorem 2 (Schinzel): Let r be a positive integer, and fix nonzero integers a 0,..., a r. Let F (x 1,..., x r ) = a r x r + + a 1 x 1 + a 0. Then there exist finite sets S and T of matrices satisfying: (i) Each matrix in S or T is an r ρ matrix with integer entries and of rank ρ for some ρ r. (ii) The matrices in S and T are computable.
12 (iii) For every set of positive integers d 1,..., d r with F (x d 1,..., x d r) not reciprocal and d 1 < d 2 < < d r, the noncyclotomic part of F (x d 1,..., x d r) is reducible if and only if there is an r ρ matrix N in S and integers v 1,..., v ρ satisfying d 1 v 1 d 2. = N v 2. d r v ρ but there is no r ρ matrix M in T with ρ < ρ and no integers v 1,..., v ρ satisfying d v 1 1 d 2. = M v 2.. d r v ρ
13 (iii) For every set of positive integers d 1,..., d r with F (x d 1,..., x d r) not reciprocal and d 1 < d 2 < < d r, the noncyclotomic part of F (x d 1,..., x d r) is reducible if and only if there is an r ρ matrix N in S and integers v 1,..., v ρ satisfying d 1 v 1 d 2. = N v 2. d r v ρ but there is no r ρ matrix M in T with ρ < ρ and no integers v 1,..., v ρ satisfying d v 1 1 d 2. = M v 2.. d r v ρ
14 (iii) For every set of positive integers d 1,..., d r with F (x d 1,..., x d r) not reciprocal and d 1 < d 2 < < d r, the noncyclotomic part of F (x d 1,..., x d r) is reducible if and only if there is an r ρ matrix N in S and integers v 1,..., v ρ satisfying d 1 v 1 d 2. = N v 2. d r v ρ but there is no r ρ matrix M in T with ρ < ρ and no integers v 1,..., v ρ satisfying d v 1 1 d 2. = M v 2.. d r v ρ
15 Theorem: There is an algorithm with the following property: Given a nonreciprocal f(x) Z[x] with N nonzero terms, degree n and height H, the algorithm determines whether f(x) is irreducible in time c(n, H)(log n) c (N) where c(n, H) depends only on N and H and c (N) depends only on N.
16 Theorem: There is an algorithm with the following property: Given a nonreciprocal f(x) Z[x] with N nonzero terms, degree n and height H, the algorithm determines whether f(x) is irreducible in time c(n, H)(log n) c (N) where c(n, H) depends only on N and H and c (N) depends only on N.
17 Proof. Let f(x) = a r x d r + + a 1 x d 1 + a 0.
18 Proof. Let f(x) = a r x d r + + a 1 x d 1 + a 0. Consider F (x 1,..., x r ) = a r x r + + a 1 x 1 + a 0 so that F (x d 1,..., x d r ) = a r x d r + + a 1 x d 1 + a 0.
19 Proof. Let Consider so that f(x) = a r x d r + + a 1 x d 1 + a 0. F (x 1,..., x r ) = a r x r + + a 1 x 1 + a 0 F (x d 1,..., x d r ) = a r x d r + + a 1 x d 1 + a 0. Begin the algorithm by constructing the finite sets S and T of matrices in Schinzel s Theorem 2.
20 Proof. Let Consider so that f(x) = a r x d r + + a 1 x d 1 + a 0. F (x 1,..., x r ) = a r x r + + a 1 x 1 + a 0 F (x d 1,..., x d r ) = a r x d r + + a 1 x d 1 + a 0. Begin the algorithm by constructing the finite sets S and T of matrices in Schinzel s Theorem 2. Observe that S and T depend on F and not on the d 1,..., d r.
21 Proof. Let Consider so that f(x) = a r x d r + + a 1 x d 1 + a 0. F (x 1,..., x r ) = a r x r + + a 1 x 1 + a 0 F (x d 1,..., x d r ) = a r x d r + + a 1 x d 1 + a 0. Begin the algorithm by constructing the finite sets S and T of matrices in Schinzel s Theorem 2. Observe that S and T depend on F and not on the d 1,..., d r, so this takes running time c 1 (N, H).
22 Next, the algorithm checks each matrix N in S to see if d 1 v 1 d 2. = N v 2. d r for some integers v 1,..., v ρ. v ρ
23 Next, the algorithm checks each matrix N in S to see if d 1 v 1 d 2. = N v 2. d r for some integers v 1,..., v ρ. In other words, v 1,..., v ρ are unknowns and elementary row operations are done to solve the above system of equations. v ρ
24 Next, the algorithm checks each matrix N in S to see if d 1 v 1 d 2. = N v 2. d r for some integers v 1,..., v ρ. In other words, v 1,..., v ρ are unknowns and elementary row operations are done to solve the above system of equations. The rank of N is ρ, so if a solution exists, then it is unique. v ρ
25 Next, the algorithm checks each matrix N in S to see if d 1 v 1 d 2. = N v 2. d r for some integers v 1,..., v ρ. In other words, v 1,..., v ρ are unknowns and elementary row operations are done to solve the above system of equations. The rank of N is ρ, so if a solution exists, then it is unique. This involves performing elementary operations (+,,, and ) with entries in N and the d j (which are n). v ρ
26 Next, the algorithm checks each matrix N in S to see if d 1 d 2. d r = N for some integers v 1,..., v ρ. In other words, v 1,..., v ρ are unknowns and elementary row operations are done to solve the above system of equations. The rank of N is ρ, so if a solution exists, then it is unique. This involves performing elementary operations (+,,, and ) with entries in N and the d j (which are n). The running time here is c 2 (N, H) log 2 n. v 1 v 2. v ρ
27 Next, the algorithm checks each matrix M in T to see if d 1 v 1 d 2. = M v 2. d r for some integers v 1,..., v ρ by using elementary row v ρ operations to solve the system of equations for the v j.
28 Next, the algorithm checks each matrix M in T to see if d 1 v 1 d 2. = M v 2. d r for some integers v 1,..., v ρ by using elementary row operations to solve the system of equations for the v j. The rank of M is ρ, so if a solution exists, then it is unique. v ρ
29 Next, the algorithm checks each matrix M in T to see if d 1 v 1 d 2. = M v 2. d r for some integers v 1,..., v ρ by using elementary row operations to solve the system of equations for the v j. The rank of M is ρ, so if a solution exists, then it is unique. This involves performing elementary operations (+,,, and ) with entries in M and the d j (which are n). v ρ
30 Next, the algorithm checks each matrix M in T to see if d 1 v 1 d 2. = M v 2. d r for some integers v 1,..., v ρ by using elementary row operations to solve the system of equations for the v j. The rank of M is ρ, so if a solution exists, then it is unique. This involves performing elementary operations (+,,, and ) with entries in M and the d j (which are n). The running time is again c 2 (N, H) log 2 n. v ρ
31 Schinzel s Theorem 2 now indicates to us whether f(x) has a reducible noncyclotomic part.
32 Schinzel s Theorem 2 now indicates to us whether f(x) has a reducible noncyclotomic part. If so, then we output that f(x) is reducible. If not, we have more work to do.
33 Recall, we had the following theorem. Theorem: There is an algorithm that has the following property: given f(x) = N j=1 a j x d j Z[x] with deg f = n, the algorithm determines whether f(x) has a cyclotomic factor and with running time exp ( (2 + o(1)) N/ log N(log N + log log n) ) log(h + 1) as N tends to infinity, where H = max 1 j N { a j }.
34 Recall, we had the following theorem. Theorem: There is an algorithm that has the following property: given f(x) = N j=1 a j x d j Z[x] with deg f = n, the algorithm determines whether f(x) has a cyclotomic factor and with running time exp ( (2 + o(1)) N/ log N(log N + log log n) ) log(h + 1) as N tends to infinity, where H = max 1 j N { a j }. We ll come back to this.
35 exp ( (2 + o(1)) N/ log N(log N + log log n) ) log(h + 1)
36 exp ( (2 + o(1)) N/ log N(log N + log log n) ) log(h + 1) split into two parts exp ( (2 + o(1)) N/ log N(log N) ) log(h + 1) exp ( (2 + o(1)) N/ log N (log log n) )
37 exp ( (2 + o(1)) N/ log N(log N + log log n) ) log(h + 1) split into two parts c 3 (N, H) exp ( (2 + o(1)) N/ log N (log log n) )
38 exp ( (2 + o(1)) N/ log N(log N + log log n) ) log(h + 1) split into two parts c 3 (N, H) (log n) c 4(N)
39 Theorem: There is an algorithm that has the following property: given f(x) = N j=1 a j x d j Z[x] with deg f = n, the algorithm determines whether f(x) has a cyclotomic factor and with running time c 3 (N, H)(log n) c 4(N) as N tends to infinity, where H = max 1 j N { a j }.
40 Theorem: There is an algorithm that has the following property: given f(x) = N j=1 a j x d j Z[x] with deg f = n, the algorithm determines whether f(x) has a cyclotomic factor and with running time c 3 (N, H)(log n) c 4(N) as N tends to infinity, where H = max 1 j N { a j }. Algorithm Continued: If the noncyclotomic part of f(x) is irreducible, then use the algorithm in the above theorem.
41 Theorem: There is an algorithm that has the following property: given f(x) = N j=1 a j x d j Z[x] with deg f = n, the algorithm determines whether f(x) has a cyclotomic factor and with running time c 3 (N, H)(log n) c 4(N) as N tends to infinity, where H = max 1 j N { a j }. Algorithm Continued: If the noncyclotomic part of f(x) is irreducible, then use the algorithm in the above theorem. This completes the proof of the theorem.
42 A CURIOUS CONNECTION WITH THE ODD COVERING PROBLEM
43 Coverings of the Integers: A covering of the integers is a system of congruences x a j (mod m j ) having the property that every integer satisfies at least one of the congruences.
44 Coverings of the Integers: A covering of the integers is a system of congruences x a j (mod m j ) having the property that every integer satisfies at least one of the congruences. Example 1: x 0 (mod 2) x 1 (mod 2)
45 Example 2: x 0 (mod 2) x 2 (mod 3) x 1 (mod 4) x 1 (mod 6) x 3 (mod 12)
46 Example 2: x 0 (mod 2) x 2 (mod 3) x 1 (mod 4) x 1 (mod 6) x 3 (mod 12)
47 Open Problem: Does there exist an odd covering of the integers, a covering consisting of distinct odd moduli > 1?
48 Open Problem: Does there exist an odd covering of the integers, a covering consisting of distinct odd moduli > 1? Erdős: $25 (for proof none exists)
49 Open Problem: Does there exist an odd covering of the integers, a covering consisting of distinct odd moduli > 1? Erdős: $25 (for proof none exists) Selfridge: $2000 (for explicit example)
50 Sierpinski s Application: There exist infinitely many (even a positive proportion of) positive integers k such that k 2 n + 1 is composite for all nonnegative integers n.
51 Sierpinski s Application: There exist infinitely many (even a positive proportion of) positive integers k such that k 2 n + 1 is composite for all nonnegative integers n. Selfridge s Example: k = (smallest odd known)
52 Sierpinski s Application: There exist infinitely many (even a positive proportion of) positive integers k such that k 2 n + 1 is composite for all nonnegative integers n. Selfridge s Example: k = (smallest odd known) Polynomial Question: Does there exist f(x) Z[x] such that f(x)x n + 1 is reducible for all nonnegative integers n?
53 Sierpinski s Application: There exist infinitely many (even a positive proportion of) positive integers k such that k 2 n + 1 is composite for all nonnegative integers n. Selfridge s Example: k = (smallest odd known) Polynomial Question: Does there exist f(x) Z[x] such that f(x)x n + 1 is reducible for all nonnegative integers n? Require: f(1) 1
54 Sierpinski s Application: There exist infinitely many (even a positive proportion of) positive integers k such that k 2 n + 1 is composite for all nonnegative integers n. Selfridge s Example: k = (smallest odd known) Polynomial Question: Does there exist f(x) Z[x] such that f(1) 1 and f(x)x n + 1 is reducible for all nonnegative integers n?
55 Sierpinski s Application: There exist infinitely many (even a positive proportion of) positive integers k such that k 2 n + 1 is composite for all nonnegative integers n. Selfridge s Example: k = (smallest odd known) Polynomial Question: Does there exist f(x) Z[x] such that f(1) 1 and f(x)x n + 1 is reducible for all nonnegative integers n? Answer: Nobody knows.
56 Schinzel s Example: (5x 9 +6x 8 +3x 6 +8x 5 +9x 3 +6x 2 +8x+3)x n +12 is reducible for all nonnegative integers n
57 Schinzel s Example: (5x 9 +6x 8 +3x 6 +8x 5 +9x 3 +6x 2 +8x+3)x n +12 is reducible for all nonnegative integers n Comment: For each n, the above polynomial is divisible by at least one of Φ k (x) where k {2, 3, 4,6, 12}.
58 Schinzel s Example: (5x 9 +6x 8 +3x 6 +8x 5 +9x 3 +6x 2 +8x+3)x n +12 is reducible for all nonnegative integers n Comment: For each n, the above polynomial is divisible by at least one of Φ k (x) where k {2, 3, 4,6, 12}. n 0 (mod 2) = f(x)x n (mod x + 1)
59 Schinzel s Example: (5x 9 +6x 8 +3x 6 +8x 5 +9x 3 +6x 2 +8x+3)x n +12 is reducible for all nonnegative integers n Comment: For each n, the above polynomial is divisible by at least one of Φ k (x) where k {2, 3, 4,6, 12}. n 0 (mod 2) = f(x)x n (mod x + 1) n 2 (mod 3) = f(x)x n (mod x 2 + x + 1)
60 Schinzel s Example: (5x 9 +6x 8 +3x 6 +8x 5 +9x 3 +6x 2 +8x+3)x n +12 is reducible for all nonnegative integers n Comment: For each n, the above polynomial is divisible by at least one of Φ k (x) where k {2, 3, 4,6, 12}. n 0 (mod 2) = f(x)x n (mod x + 1) n 2 (mod 3) = f(x)x n (mod x 2 + x + 1) n 1 (mod 4) = f(x)x n (mod x 2 + 1)
61 Schinzel s Example: (5x 9 +6x 8 +3x 6 +8x 5 +9x 3 +6x 2 +8x+3)x n +12 is reducible for all nonnegative integers n Comment: For each n, the above polynomial is divisible by at least one of Φ k (x) where k {2, 3, 4,6, 12}. n 0 (mod 2) = f(x)x n (mod x + 1) n 2 (mod 3) = f(x)x n (mod x 2 + x + 1) n 1 (mod 4) = f(x)x n (mod x 2 + 1) n 1 (mod 6) = f(x)x n (mod x 2 x + 1)
62 Schinzel s Example: (5x 9 +6x 8 +3x 6 +8x 5 +9x 3 +6x 2 +8x+3)x n +12 is reducible for all nonnegative integers n Comment: For each n, the above polynomial is divisible by at least one of Φ k (x) where k {2, 3, 4,6, 12}. n 0 (mod 2) = f(x)x n (mod x + 1) n 2 (mod 3) = f(x)x n (mod x 2 + x + 1) n 1 (mod 4) = f(x)x n (mod x 2 + 1) n 1 (mod 6) = f(x)x n (mod x 2 x + 1) n 3 (mod 12) = f(x)x n (mod x 4 x 2 + 1)
63 Schinzel s Example: (5x 9 +6x 8 +3x 6 +8x 5 +9x 3 +6x 2 +8x+3)x n +12 is reducible for all nonnegative integers n Theorem. There exists an f(x) Z[x] with nonnegative coefficients such that f(x)x n +4 is reducible for all nonnegative integers n.
64 Schinzel s Example: (5x 9 +6x 8 +3x 6 +8x 5 +9x 3 +6x 2 +8x+3)x n +12 is reducible for all nonnegative integers n Theorem. There exists an f(x) Z[x] with nonnegative coefficients such that f(x)x n +4 is reducible for all nonnegative integers n.
65 Schinzel s Example: (5x 9 +6x 8 +3x 6 +8x 5 +9x 3 +6x 2 +8x+3)x n +12 is reducible for all nonnegative integers n Theorem. There exists an f(x) Z[x] with nonnegative coefficients such that f(x)x n +4 is reducible for all nonnegative integers n.
66 Schinzel s Example: (5x 9 +6x 8 +3x 6 +8x 5 +9x 3 +6x 2 +8x+3)x n +12 is reducible for all nonnegative integers n Theorem. There exists an f(x) Z[x] with nonnegative coefficients such that f(x)x n +4 is reducible for all nonnegative integers n. Comment: For each n, the first polynomial is divisible by at least one Φ k (x) where k divides 12.
67 Schinzel s Example: (5x 9 +6x 8 +3x 6 +8x 5 +9x 3 +6x 2 +8x+3)x n +12 is reducible for all nonnegative integers n Theorem. There exists an f(x) Z[x] with nonnegative coefficients such that f(x)x n +4 is reducible for all nonnegative integers n. Comment: For each n, the second polynomial is divisible by at least one Φ k (x) where k divides some integer N having more than digits.
68 Schinzel s Example: (5x 9 +6x 8 +3x 6 +8x 5 +9x 3 +6x 2 +8x+3)x n +12 is reducible for all nonnegative integers n Theorem. There exists an f(x) Z[x] with nonnegative coefficients such that f(x)x n +4 is reducible for all nonnegative integers n. Comment: For each n, the second polynomial is divisible by at least one Φ k (x) where k divides
69 Schinzel s Theorem: If there is an f(x) Z[x] such that f(1) 1 and f(x)x n + 1 is reducible for all nonnegative integers n, then there is an odd covering of the integers.
70 TURÁN S CONJECTURE
71 Conjecture: There is an absolute constant C such that if r f(x) = a j x j Z[x], then there is a g(x) = j=0 r b j x j Z[x] j=0 irreducible (over Q) such that r b j a j C. j=0
72 Conjecture: There is an absolute constant C such that if r f(x) = a j x j Z[x], then there is a g(x) = j=0 r j=0 irreducible (over Q) such that b j x j Z[x] r j=0 b j a j C. Comment: The conjecture remains open. If we take g(x) = s j=0 b j x j Z[x] where possibly s > r, then the problem has been resolved by Schinzel.
73 First Attack on Turán s Problem: Old Theorem: When m is large, either u(x)x m + v(x) has an obvious factorization or the nonreciprocal part of u(x)x m + v(x) is irreducible.
74 First Attack on Turán s Problem: Old Theorem: When m is large, either u(x)x m + v(x) has an obvious factorization or the nonreciprocal part of u(x)x m + v(x) is irreducible. Idea: Consider g(x) = x n + f(x). If one can show g(x) is irreducible for some n, then the conjecture of Turán (modified so deg g > deg f is allowed) is resolved with C = 1.
75 First Attack on Turán s Problem: Old Theorem: When m is large, either u(x)x m + v(x) has an obvious factorization or the nonreciprocal part of u(x)x m + v(x) is irreducible. Idea: Consider g(x) = x n + f(x). If f(0) = 0 or f(1) = 1, then consider instead g(x) = x n + f(x) ± 1. If one can show g(x) is irreducible for some n, then the conjecture of Turán (modified so deg g > deg f is allowed) is resolved with C = 2.
76 First Attack on Turán s Problem: Idea: Consider g(x) = x n + f(x). If f(0) = 0 or f(1) = 1, then consider instead g(x) = x n + f(x) ± 1. If one can show g(x) is irreducible for some n, then the conjecture of Turán (modified so deg g > deg f is allowed) is resolved with C = 2.
77 First Attack on Turán s Problem: Idea: Consider g(x) = x n + f(x). If f(0) = 0 or f(1) = 1, then consider instead g(x) = x n + f(x) ± 1. If one can show g(x) is irreducible for some n, then the conjecture of Turán (modified so deg g > deg f is allowed) is resolved with C = 2. Problem: Dealing with g(x) = x n + f(x) is essentially equivalent to the odd covering problem. So this is hard.
78 Second Attack on Turán s Problem: Idea: Consider g(x) = x m ± x n + f(x). If f(0) = 0, then consider instead g(x) = x m ± x n + f(x) ± 1.
79 Theorem (Schinzel): For every r f(x) = a j x j Z[x], j=0 there exist infinitely many irreducible s g(x) = b j x j Z[x] such that max{r,s} j=0 j=0 a j b j { 2 if f(0) 0 3 always.
80 Theorem (Schinzel): For every r f(x) = a j x j Z[x], j=0 there exist infinitely many irreducible s g(x) = b j x j Z[x] such that max{r,s} j=0 j=0 a j b j One of these is such that { 2 if f(0) 0 3 always. s < exp ( (5r + 7)( f 2 + 3) ).
81 Ideas Behind Proof:
82 Ideas Behind Proof: Consider F (x) = x m + x n + f(x) with m (M, 2M] and n (N, 2N] where M and N are large and M > N.
83 Ideas Behind Proof: Consider F (x) = x m + x n + f(x) with m (M, 2M] and n (N, 2N] where M and N are large and M > N. Apply result on u(x)x m + v(x) with u(x) = 1 and v(x) = x n + f(x) to reduce problem to consideration of reciprocal factors.
84 Ideas Behind Proof: Consider F (x) = x m + x n + f(x) with m (M, 2M] and n (N, 2N] where M and N are large and M > N. Apply result on u(x)x m + v(x) with u(x) = 1 and v(x) = x n + f(x) to reduce problem to consideration of reciprocal factors. Find a bound on the number of x m + x n + f(x) with reciprocal noncyclotomic factors.
85 Ideas Behind Proof: To bound the x m + x n + f(x) with cyclotomic factors, set A = {(m, n) : M <m 2M, N <n 2N}, and let A p A (arising from when F (ζ p k) = 0). Use a sieve argument to estimate the size of A A p.
86 Ideas Behind Proof: To bound the x m + x n + f(x) with cyclotomic factors, set A = {(m, n) : M <m 2M, N <n 2N}, and let A p A (arising from when F (ζ p k) = 0). Use a sieve argument to estimate the size of A A p. Deduce that some F (x) = x m +x n +f(x) with m (M, 2M] and n (N, 2N] is irreducible (where M and N are large and M > N).
87 Current Knowledge: Theorem: Given f(x) = r j=0 infinitely many irreducible g(x) = such that max{r,s} j=0 One of these is such that a j x j Z[x], there are s j=0 a j b j 5. s 4r exp ( 4 f ). b j x j Z[x]
88 Current Knowledge: Theorem: Given f(x) = r j=0 infinitely many irreducible g(x) = such that max{r,s} j=0 One of these is such that a j x j Z[x], there are s j=0 a j b j 5. s 4r exp ( 4 f ). b j x j Z[x]
89 Current Knowledge: Theorem: Given f(x) = r j=0 infinitely many irreducible g(x) = such that max{r,s} j=0 One of these is such that a j x j Z[x], there are s j=0 a j b j 3. s some polynomial in r. b j x j Z[x]
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