FACTORING SPARSE POLYNOMIALS
|
|
- Amos Warren
- 8 years ago
- Views:
Transcription
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]
Introduction to Finite Fields (cont.)
Chapter 6 Introduction to Finite Fields (cont.) 6.1 Recall Theorem. Z m is a field m is a prime number. Theorem (Subfield Isomorphic to Z p ). Every finite field has the order of a power of a prime number
More informationMOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao yufeiz@mit.edu
Integer Polynomials June 9, 007 Yufei Zhao yufeiz@mit.edu We will use Z[x] to denote the ring of polynomials with integer coefficients. We begin by summarizing some of the common approaches used in dealing
More informationminimal polyonomial Example
Minimal Polynomials Definition Let α be an element in GF(p e ). We call the monic polynomial of smallest degree which has coefficients in GF(p) and α as a root, the minimal polyonomial of α. Example: We
More informationCHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY
January 10, 2010 CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY The set of polynomials over a field F is a ring, whose structure shares with the ring of integers many characteristics.
More informationModule MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013
Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013 D. R. Wilkins Copyright c David R. Wilkins 1997 2013 Contents A Cyclotomic Polynomials 79 A.1 Minimum Polynomials of Roots of
More informationDie ganzen zahlen hat Gott gemacht
Die ganzen zahlen hat Gott gemacht Polynomials with integer values B.Sury A quote attributed to the famous mathematician L.Kronecker is Die Ganzen Zahlen hat Gott gemacht, alles andere ist Menschenwerk.
More informationIrreducibility criteria for compositions and multiplicative convolutions of polynomials with integer coefficients
DOI: 10.2478/auom-2014-0007 An. Şt. Univ. Ovidius Constanţa Vol. 221),2014, 73 84 Irreducibility criteria for compositions and multiplicative convolutions of polynomials with integer coefficients Anca
More informationPrime Numbers and Irreducible Polynomials
Prime Numbers and Irreducible Polynomials M. Ram Murty The similarity between prime numbers and irreducible polynomials has been a dominant theme in the development of number theory and algebraic geometry.
More informationFactoring Polynomials
Factoring Polynomials Sue Geller June 19, 2006 Factoring polynomials over the rational numbers, real numbers, and complex numbers has long been a standard topic of high school algebra. With the advent
More informationArithmetic algorithms for cryptology 5 October 2015, Paris. Sieves. Razvan Barbulescu CNRS and IMJ-PRG. R. Barbulescu Sieves 0 / 28
Arithmetic algorithms for cryptology 5 October 2015, Paris Sieves Razvan Barbulescu CNRS and IMJ-PRG R. Barbulescu Sieves 0 / 28 Starting point Notations q prime g a generator of (F q ) X a (secret) integer
More information7. Some irreducible polynomials
7. Some irreducible polynomials 7.1 Irreducibles over a finite field 7.2 Worked examples Linear factors x α of a polynomial P (x) with coefficients in a field k correspond precisely to roots α k [1] of
More informationMethods for Finding Bases
Methods for Finding Bases Bases for the subspaces of a matrix Row-reduction methods can be used to find bases. Let us now look at an example illustrating how to obtain bases for the row space, null space,
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
More informationPrime numbers and prime polynomials. Paul Pollack Dartmouth College
Prime numbers and prime polynomials Paul Pollack Dartmouth College May 1, 2008 Analogies everywhere! Analogies in elementary number theory (continued fractions, quadratic reciprocity, Fermat s last theorem)
More informationPROBLEM SET 6: POLYNOMIALS
PROBLEM SET 6: POLYNOMIALS 1. introduction In this problem set we will consider polynomials with coefficients in K, where K is the real numbers R, the complex numbers C, the rational numbers Q or any other
More informationPUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.
PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include
More informationit is easy to see that α = a
21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore
More informationWinter Camp 2011 Polynomials Alexander Remorov. Polynomials. Alexander Remorov alexanderrem@gmail.com
Polynomials Alexander Remorov alexanderrem@gmail.com Warm-up Problem 1: Let f(x) be a quadratic polynomial. Prove that there exist quadratic polynomials g(x) and h(x) such that f(x)f(x + 1) = g(h(x)).
More informationChapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.
Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize
More information3 1. Note that all cubes solve it; therefore, there are no more
Math 13 Problem set 5 Artin 11.4.7 Factor the following polynomials into irreducible factors in Q[x]: (a) x 3 3x (b) x 3 3x + (c) x 9 6x 6 + 9x 3 3 Solution: The first two polynomials are cubics, so if
More informationFaster deterministic integer factorisation
David Harvey (joint work with Edgar Costa, NYU) University of New South Wales 25th October 2011 The obvious mathematical breakthrough would be the development of an easy way to factor large prime numbers
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationHomework until Test #2
MATH31: Number Theory Homework until Test # Philipp BRAUN Section 3.1 page 43, 1. It has been conjectured that there are infinitely many primes of the form n. Exhibit five such primes. Solution. Five such
More informationU.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra
U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory
More informationFACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z
FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization
More informationReal Roots of Univariate Polynomials with Real Coefficients
Real Roots of Univariate Polynomials with Real Coefficients mostly written by Christina Hewitt March 22, 2012 1 Introduction Polynomial equations are used throughout mathematics. When solving polynomials
More informationOn the number-theoretic functions ν(n) and Ω(n)
ACTA ARITHMETICA LXXVIII.1 (1996) On the number-theoretic functions ν(n) and Ω(n) by Jiahai Kan (Nanjing) 1. Introduction. Let d(n) denote the divisor function, ν(n) the number of distinct prime factors,
More informationI. GROUPS: BASIC DEFINITIONS AND EXAMPLES
I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called
More informationReduced echelon form: Add the following conditions to conditions 1, 2, and 3 above:
Section 1.2: Row Reduction and Echelon Forms Echelon form (or row echelon form): 1. All nonzero rows are above any rows of all zeros. 2. Each leading entry (i.e. left most nonzero entry) of a row is in
More informationThe Division Algorithm for Polynomials Handout Monday March 5, 2012
The Division Algorithm for Polynomials Handout Monday March 5, 0 Let F be a field (such as R, Q, C, or F p for some prime p. This will allow us to divide by any nonzero scalar. (For some of the following,
More informationMA4001 Engineering Mathematics 1 Lecture 10 Limits and Continuity
MA4001 Engineering Mathematics 1 Lecture 10 Limits and Dr. Sarah Mitchell Autumn 2014 Infinite limits If f(x) grows arbitrarily large as x a we say that f(x) has an infinite limit. Example: f(x) = 1 x
More informationAlex, I will take congruent numbers for one million dollars please
Alex, I will take congruent numbers for one million dollars please Jim L. Brown The Ohio State University Columbus, OH 4310 jimlb@math.ohio-state.edu One of the most alluring aspectives of number theory
More informationBX in ( u, v) basis in two ways. On the one hand, AN = u+
1. Let f(x) = 1 x +1. Find f (6) () (the value of the sixth derivative of the function f(x) at zero). Answer: 7. We expand the given function into a Taylor series at the point x = : f(x) = 1 x + x 4 x
More informationThe cyclotomic polynomials
The cyclotomic polynomials Notes by G.J.O. Jameson 1. The definition and general results We use the notation e(t) = e 2πit. Note that e(n) = 1 for integers n, e(s + t) = e(s)e(t) for all s, t. e( 1 ) =
More informationTHE NUMBER OF REPRESENTATIONS OF n OF THE FORM n = x 2 2 y, x > 0, y 0
THE NUMBER OF REPRESENTATIONS OF n OF THE FORM n = x 2 2 y, x > 0, y 0 RICHARD J. MATHAR Abstract. We count solutions to the Ramanujan-Nagell equation 2 y +n = x 2 for fixed positive n. The computational
More information8 Primes and Modular Arithmetic
8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.
More information1 Lecture: Integration of rational functions by decomposition
Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.
More informationHow To Prove The Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if
More informationNotes on Factoring. MA 206 Kurt Bryan
The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor
More informationPrimality - Factorization
Primality - Factorization Christophe Ritzenthaler November 9, 2009 1 Prime and factorization Definition 1.1. An integer p > 1 is called a prime number (nombre premier) if it has only 1 and p as divisors.
More informationa 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
More informationOn the irreducibility of certain polynomials with coefficients as products of terms in an arithmetic progression
On the irreducibility of certain polynomials with coefficients as products of terms in an arithmetic progression Carrie E. Finch and N. Saradha Abstract We prove the irreducibility of ceratin polynomials
More informationArithmetic and Algebra of Matrices
Arithmetic and Algebra of Matrices Math 572: Algebra for Middle School Teachers The University of Montana 1 The Real Numbers 2 Classroom Connection: Systems of Linear Equations 3 Rational Numbers 4 Irrational
More information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More informationFactorization Algorithms for Polynomials over Finite Fields
Degree Project Factorization Algorithms for Polynomials over Finite Fields Sajid Hanif, Muhammad Imran 2011-05-03 Subject: Mathematics Level: Master Course code: 4MA11E Abstract Integer factorization is
More informationMonogenic Fields and Power Bases Michael Decker 12/07/07
Monogenic Fields and Power Bases Michael Decker 12/07/07 1 Introduction Let K be a number field of degree k and O K its ring of integers Then considering O K as a Z-module, the nicest possible case is
More informationQuotient Rings and Field Extensions
Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.
More informationFIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper.
FIRST YEAR CALCULUS WWLCHENW L c WWWL W L Chen, 1982, 2008. 2006. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It It is is
More informationSome applications of LLL
Some applications of LLL a. Factorization of polynomials As the title Factoring polynomials with rational coefficients of the original paper in which the LLL algorithm was first published (Mathematische
More informationFactoring & Primality
Factoring & Primality Lecturer: Dimitris Papadopoulos In this lecture we will discuss the problem of integer factorization and primality testing, two problems that have been the focus of a great amount
More informationRevised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)
Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.
More information11 Ideals. 11.1 Revisiting Z
11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(
More informationThe van Hoeij Algorithm for Factoring Polynomials
The van Hoeij Algorithm for Factoring Polynomials Jürgen Klüners Abstract In this survey we report about a new algorithm for factoring polynomials due to Mark van Hoeij. The main idea is that the combinatorial
More informationLecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs
CSE599s: Extremal Combinatorics November 21, 2011 Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs Lecturer: Anup Rao 1 An Arithmetic Circuit Lower Bound An arithmetic circuit is just like
More informationOSTROWSKI FOR NUMBER FIELDS
OSTROWSKI FOR NUMBER FIELDS KEITH CONRAD Ostrowski classified the nontrivial absolute values on Q: up to equivalence, they are the usual (archimedean) absolute value and the p-adic absolute values for
More informationThe finite field with 2 elements The simplest finite field is
The finite field with 2 elements The simplest finite field is GF (2) = F 2 = {0, 1} = Z/2 It has addition and multiplication + and defined to be 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 0 0 0 = 0 0 1 = 0
More informationH/wk 13, Solutions to selected problems
H/wk 13, Solutions to selected problems Ch. 4.1, Problem 5 (a) Find the number of roots of x x in Z 4, Z Z, any integral domain, Z 6. (b) Find a commutative ring in which x x has infinitely many roots.
More informationMATH 289 PROBLEM SET 4: NUMBER THEORY
MATH 289 PROBLEM SET 4: NUMBER THEORY 1. The greatest common divisor If d and n are integers, then we say that d divides n if and only if there exists an integer q such that n = qd. Notice that if d divides
More informationON HENSEL S ROOTS AND A FACTORIZATION FORMULA IN Z[[X]]
#A47 INTEGERS 4 (204) ON HENSEL S ROOTS AND A FACTORIZATION FORMULA IN Z[[X]] Daniel Birmaer Department of Mathematics, Nazareth College, Rochester, New Yor abirma6@naz.edu Juan B. Gil Penn State Altoona,
More informationHYPERELLIPTIC CURVE METHOD FOR FACTORING INTEGERS. 1. Thoery and Algorithm
HYPERELLIPTIC CURVE METHOD FOR FACTORING INTEGERS WENHAN WANG 1. Thoery and Algorithm The idea of the method using hyperelliptic curves to factor integers is similar to the elliptic curve factoring method.
More informationFactoring of Prime Ideals in Extensions
Chapter 4 Factoring of Prime Ideals in Extensions 4. Lifting of Prime Ideals Recall the basic AKLB setup: A is a Dedekind domain with fraction field K, L is a finite, separable extension of K of degree
More informationNew Methods Providing High Degree Polynomials with Small Mahler Measure
New Methods Providing High Degree Polynomials with Small Mahler Measure G. Rhin and J.-M. Sac-Épée CONTENTS 1. Introduction 2. A Statistical Method 3. A Minimization Method 4. Conclusion and Prospects
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column
More informationLecture 6: Finite Fields (PART 3) PART 3: Polynomial Arithmetic. Theoretical Underpinnings of Modern Cryptography
Lecture 6: Finite Fields (PART 3) PART 3: Polynomial Arithmetic Theoretical Underpinnings of Modern Cryptography Lecture Notes on Computer and Network Security by Avi Kak (kak@purdue.edu) January 29, 2015
More informationr + s = i + j (q + t)n; 2 rs = ij (qj + ti)n + qtn.
Chapter 7 Introduction to finite fields This chapter provides an introduction to several kinds of abstract algebraic structures, particularly groups, fields, and polynomials. Our primary interest is in
More informationChapter 1. Search for Good Linear Codes in the Class of Quasi-Cyclic and Related Codes
Chapter 1 Search for Good Linear Codes in the Class of Quasi-Cyclic and Related Codes Nuh Aydin and Tsvetan Asamov Department of Mathematics, Kenyon College Gambier, OH, USA 43022 {aydinn,asamovt}@kenyon.edu
More informationFACTORING AFTER DEDEKIND
FACTORING AFTER DEDEKIND KEITH CONRAD Let K be a number field and p be a prime number. When we factor (p) = po K into prime ideals, say (p) = p e 1 1 peg g, we refer to the data of the e i s, the exponents
More informationChapter 13: Basic ring theory
Chapter 3: Basic ring theory Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 3: Basic ring
More informationLecture 13 - Basic Number Theory.
Lecture 13 - Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are non-negative integers. We say that A divides B, denoted
More informationON GALOIS REALIZATIONS OF THE 2-COVERABLE SYMMETRIC AND ALTERNATING GROUPS
ON GALOIS REALIZATIONS OF THE 2-COVERABLE SYMMETRIC AND ALTERNATING GROUPS DANIEL RABAYEV AND JACK SONN Abstract. Let f(x) be a monic polynomial in Z[x] with no rational roots but with roots in Q p for
More informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More informationGalois Theory, First Edition
Galois Theory, First Edition David A. Cox, published by John Wiley & Sons, 2004 Errata as of September 18, 2012 This errata sheet is organized by which printing of the book you have. The printing can be
More informationSouth Carolina College- and Career-Ready (SCCCR) Pre-Calculus
South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More informationHOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!
Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following
More informationLecture 3: Finding integer solutions to systems of linear equations
Lecture 3: Finding integer solutions to systems of linear equations Algorithmic Number Theory (Fall 2014) Rutgers University Swastik Kopparty Scribe: Abhishek Bhrushundi 1 Overview The goal of this lecture
More informationFactoring pq 2 with Quadratic Forms: Nice Cryptanalyses
Factoring pq 2 with Quadratic Forms: Nice Cryptanalyses Phong Nguyễn http://www.di.ens.fr/~pnguyen & ASIACRYPT 2009 Joint work with G. Castagnos, A. Joux and F. Laguillaumie Summary Factoring A New Factoring
More informationNumber Theory Hungarian Style. Cameron Byerley s interpretation of Csaba Szabó s lectures
Number Theory Hungarian Style Cameron Byerley s interpretation of Csaba Szabó s lectures August 20, 2005 2 0.1 introduction Number theory is a beautiful subject and even cooler when you learn about it
More information9. POLYNOMIALS. Example 1: The expression a(x) = x 3 4x 2 + 7x 11 is a polynomial in x. The coefficients of a(x) are the numbers 1, 4, 7, 11.
9. POLYNOMIALS 9.1. Definition of a Polynomial A polynomial is an expression of the form: a(x) = a n x n + a n-1 x n-1 +... + a 1 x + a 0. The symbol x is called an indeterminate and simply plays the role
More informationZeros of Polynomial Functions
Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of
More informationSUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by
SUBGROUPS OF CYCLIC GROUPS KEITH CONRAD 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by g = {g k : k Z}. If G = g, then G itself is cyclic, with g as a generator. Examples
More informationInteger roots of quadratic and cubic polynomials with integer coefficients
Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street
More information1 = (a 0 + b 0 α) 2 + + (a m 1 + b m 1 α) 2. for certain elements a 0,..., a m 1, b 0,..., b m 1 of F. Multiplying out, we obtain
Notes on real-closed fields These notes develop the algebraic background needed to understand the model theory of real-closed fields. To understand these notes, a standard graduate course in algebra is
More informationSome Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.
Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,
More informationA New Generic Digital Signature Algorithm
Groups Complex. Cryptol.? (????), 1 16 DOI 10.1515/GCC.????.??? de Gruyter???? A New Generic Digital Signature Algorithm Jennifer Seberry, Vinhbuu To and Dongvu Tonien Abstract. In this paper, we study
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationCollinear Points in Permutations
Collinear Points in Permutations Joshua N. Cooper Courant Institute of Mathematics New York University, New York, NY József Solymosi Department of Mathematics University of British Columbia, Vancouver,
More informationLecture Notes on Polynomials
Lecture Notes on Polynomials Arne Jensen Department of Mathematical Sciences Aalborg University c 008 Introduction These lecture notes give a very short introduction to polynomials with real and complex
More informationSolving Linear Systems, Continued and The Inverse of a Matrix
, Continued and The of a Matrix Calculus III Summer 2013, Session II Monday, July 15, 2013 Agenda 1. The rank of a matrix 2. The inverse of a square matrix Gaussian Gaussian solves a linear system by reducing
More informationHow To Know If A Domain Is Unique In An Octempo (Euclidean) Or Not (Ecl)
Subsets of Euclidean domains possessing a unique division algorithm Andrew D. Lewis 2009/03/16 Abstract Subsets of a Euclidean domain are characterised with the following objectives: (1) ensuring uniqueness
More informationThe sum of digits of polynomial values in arithmetic progressions
The sum of digits of polynomial values in arithmetic progressions Thomas Stoll Institut de Mathématiques de Luminy, Université de la Méditerranée, 13288 Marseille Cedex 9, France E-mail: stoll@iml.univ-mrs.fr
More informationI. Introduction. MPRI Cours 2-12-2. Lecture IV: Integer factorization. What is the factorization of a random number? II. Smoothness testing. F.
F. Morain École polytechnique MPRI cours 2-12-2 2013-2014 3/22 F. Morain École polytechnique MPRI cours 2-12-2 2013-2014 4/22 MPRI Cours 2-12-2 I. Introduction Input: an integer N; logox F. Morain logocnrs
More informationFactoring Algorithms
Factoring Algorithms The p 1 Method and Quadratic Sieve November 17, 2008 () Factoring Algorithms November 17, 2008 1 / 12 Fermat s factoring method Fermat made the observation that if n has two factors
More informationChapter 3. if 2 a i then location: = i. Page 40
Chapter 3 1. Describe an algorithm that takes a list of n integers a 1,a 2,,a n and finds the number of integers each greater than five in the list. Ans: procedure greaterthanfive(a 1,,a n : integers)
More informationVector and Matrix Norms
Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a non-empty
More informationZeros of a Polynomial Function
Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we
More informationMATH PROBLEMS, WITH SOLUTIONS
MATH PROBLEMS, WITH SOLUTIONS OVIDIU MUNTEANU These are free online notes that I wrote to assist students that wish to test their math skills with some problems that go beyond the usual curriculum. These
More informationMATH 425, PRACTICE FINAL EXAM SOLUTIONS.
MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator
More informationNotes 11: List Decoding Folded Reed-Solomon Codes
Introduction to Coding Theory CMU: Spring 2010 Notes 11: List Decoding Folded Reed-Solomon Codes April 2010 Lecturer: Venkatesan Guruswami Scribe: Venkatesan Guruswami At the end of the previous notes,
More informationDiscrete Mathematics: Homework 7 solution. Due: 2011.6.03
EE 2060 Discrete Mathematics spring 2011 Discrete Mathematics: Homework 7 solution Due: 2011.6.03 1. Let a n = 2 n + 5 3 n for n = 0, 1, 2,... (a) (2%) Find a 0, a 1, a 2, a 3 and a 4. (b) (2%) Show that
More informationCryptography and Network Security. Prof. D. Mukhopadhyay. Department of Computer Science and Engineering. Indian Institute of Technology, Kharagpur
Cryptography and Network Security Prof. D. Mukhopadhyay Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur Module No. # 01 Lecture No. # 12 Block Cipher Standards
More information