# FACTORING SPARSE POLYNOMIALS

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1 FACTORING SPARSE POLYNOMIALS

2 Theorem 1 (Schinzel): Let r be a positive integer, and fix non-zero 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 non-zero 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 non-zero 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 non-reciprocal 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 non-reciprocal part of F (x d 1,..., x d r) is reducible

7 the non-reciprocal 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 non-reciprocal 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 non-zero 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 non-zero 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 non-zero 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 non-cyclotomic 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 non-cyclotomic 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 non-cyclotomic 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 non-reciprocal 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 non-reciprocal 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 non-cyclotomic part.

32 Schinzel s Theorem 2 now indicates to us whether f(x) has a reducible non-cyclotomic 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 non-cyclotomic 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 non-cyclotomic 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 non-negative 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 non-negative 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 non-negative 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 non-negative 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 non-negative 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 non-negative 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 non-negative 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 non-negative 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 non-negative 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 non-negative 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 non-negative 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 non-negative 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 non-negative 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 non-negative 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 non-negative 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 non-negative 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 non-negative 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 non-negative integers n Theorem. There exists an f(x) Z[x] with non-negative 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 non-negative integers n Theorem. There exists an f(x) Z[x] with non-negative 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 non-negative integers n Theorem. There exists an f(x) Z[x] with non-negative 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 non-negative integers n Theorem. There exists an f(x) Z[x] with non-negative 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 non-negative integers n Theorem. There exists an f(x) Z[x] with non-negative 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 non-negative integers n Theorem. There exists an f(x) Z[x] with non-negative 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 non-negative 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 non-reciprocal 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 non-reciprocal 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 non-reciprocal 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 non-cyclotomic 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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