Nemo/Hecke: Computer Algebra and Number
... [23], as taught to us by Bernard Parisse. The most important of these is cheap removal of the content that accumulates during the subresultant algorithm by analysing the input polynomials to see what form the leading coefficient of the GCD can take. We also use heuristics to determine which permutat ...
... [23], as taught to us by Bernard Parisse. The most important of these is cheap removal of the content that accumulates during the subresultant algorithm by analysing the input polynomials to see what form the leading coefficient of the GCD can take. We also use heuristics to determine which permutat ...
THE PROBABILITY OF RELATIVELY PRIME POLYNOMIALS
... Proof. We first note that a linear factor of g(x) must have the form u(x − r), where u, r ∈ Zpk , u is a unit, and r is a root of g. Therefore, the elements h(x) ∈ Ag are exactly the polynomials h(x) = α(x − r), for some α ∈ Zpk and some root r ∈ Zpk of g. Hence, to calculate |Ag |, we need to count ...
... Proof. We first note that a linear factor of g(x) must have the form u(x − r), where u, r ∈ Zpk , u is a unit, and r is a root of g. Therefore, the elements h(x) ∈ Ag are exactly the polynomials h(x) = α(x − r), for some α ∈ Zpk and some root r ∈ Zpk of g. Hence, to calculate |Ag |, we need to count ...
What We Need to Know about Rings and Modules
... Moreover we have the following uniqueness. If a = q1 q2 · · · qm is anther expression of a as a product of primes, then m = n and after a reordering of q1 , q2 , . . . , qn there are units u1 , u2 , . . . , un so that qi = ui pi for i = 1, . . . , n. Problem 9 Prove this by induction on δ(a) in the ...
... Moreover we have the following uniqueness. If a = q1 q2 · · · qm is anther expression of a as a product of primes, then m = n and after a reordering of q1 , q2 , . . . , qn there are units u1 , u2 , . . . , un so that qi = ui pi for i = 1, . . . , n. Problem 9 Prove this by induction on δ(a) in the ...
Short Programs for functions on Curves
... We further exploit the fact that there is a one-to-one correspondence between effective divisors and integral ideals of the ring of functions whose only poles are at Q. We represent each ideal by means of it Grobner Basis. In this way we avoid the necessity for root finding and factorization. In the ...
... We further exploit the fact that there is a one-to-one correspondence between effective divisors and integral ideals of the ring of functions whose only poles are at Q. We represent each ideal by means of it Grobner Basis. In this way we avoid the necessity for root finding and factorization. In the ...
Solutions - Technische Universität München
... Let υP,Q be the sign changes at −∞ minus the sign changes at ∞. Let c>0 be the number of roots of P where Q(x) > 0 and c<0 be the number of roots of P where Q(x) < 0, then υP,Q = c>0 − c<0 Then build the Sturm sequence for P = x3 + 3x2 − 4x − 12 and Q2 = (−x)2 : P0 = P = x3 + 3x2 − 4x − 12 P1 = P 0 ...
... Let υP,Q be the sign changes at −∞ minus the sign changes at ∞. Let c>0 be the number of roots of P where Q(x) > 0 and c<0 be the number of roots of P where Q(x) < 0, then υP,Q = c>0 − c<0 Then build the Sturm sequence for P = x3 + 3x2 − 4x − 12 and Q2 = (−x)2 : P0 = P = x3 + 3x2 − 4x − 12 P1 = P 0 ...
CHAP12 Polynomial Codes
... corrupt the text. Of course we could simply repeat the message. Wherever the two copies differ, the receiver will know that an error has occurred. But the receiver won't know, for each error, which version is correct. We’d need to transmit the message three times for the receiver to be able to corre ...
... corrupt the text. Of course we could simply repeat the message. Wherever the two copies differ, the receiver will know that an error has occurred. But the receiver won't know, for each error, which version is correct. We’d need to transmit the message three times for the receiver to be able to corre ...
Modeling and analyzing finite state automata in the
... 3. There exists an identity element e such that for all a ∈ G, a ∗ e = e ∗ a = a. 4. There exists an inverse element a−1 ∈ G for each a ∈ G such that a ∗ a−1 = a−1 ∗ a = e. Moreover, a group is commutative (or abelian) if for all a, b ∈ G, a ∗ b = b ∗ a. A group is called finite if the set G contain ...
... 3. There exists an identity element e such that for all a ∈ G, a ∗ e = e ∗ a = a. 4. There exists an inverse element a−1 ∈ G for each a ∈ G such that a ∗ a−1 = a−1 ∗ a = e. Moreover, a group is commutative (or abelian) if for all a, b ∈ G, a ∗ b = b ∗ a. A group is called finite if the set G contain ...
Finite fields - CSE
... In the hidden subgroup problem, we are given a group G and a function f : G → R which hides a subgroup H. By hiding a subgroup means that the functions assign the same value to two elements from the same coset and different values to elements from a different coset. The subgroup H is not known and t ...
... In the hidden subgroup problem, we are given a group G and a function f : G → R which hides a subgroup H. By hiding a subgroup means that the functions assign the same value to two elements from the same coset and different values to elements from a different coset. The subgroup H is not known and t ...
