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I'm new to this Wikipedia thing, so I'll just post my opinion here before editing. Hope that someone with more experience could help.
The point is that there is no formal (mathematically correct) definition given. What you have is just an explanation/illustration.
Generally a polynomial is defined as a sequence of elements (called coefficients) from a ring indexed by natural numbers with the following characteristic: there is a natural number n (called the degree) so that the n-th coefficient is non-zero and all coefficients with index higher than are zero.
i.e. it is something like P={a0,a1,...,an,0,0,...} where a0,...an are ellements of R and 0 is the zero in R.
More formally: a polynomial is a function P from N (natural numbers) to R (where R is a ring) where there is n in N (the degree) so that P(n) is not zero and for all m>n P(m)=0.
The thing is that the definition of the degree is embedded in the definition of the polynomial. What would be the best way to include here the formal/mathematically correct definition of the degree - should we repeat the polynomial definition here? - AdamSmithee 22:25, 17 November 2005 (UTC)
Two conflicting definitions seem to be given. The first paragraph says the degree is the sum of the powers of all terms, the second says it is the sum of powers in one term. That is confusing. Jewels Vern ( talk) 05:23, 18 October 2013 (UTC)
I've had a bit of a debate with someone regarding degrees 0 and 1 of a polynomial. My argument is this: the degree of is 1, as the exponent on the variable is 1. But the degree of any constant term is 0. E.g: , as 0 is the exponent on x. To my surprise, they were arguing that , therefore the degree is 1. Surely the coefficients can't be used in determining the degree, otherwise the degree of would be 3?! I'm not a mathematician so I'm a bit reluctant to change the article, but would it be reasonable to clarify this in the article? Or maybe I was wrong? -- 146.227.11.232 15:55, 13 January 2006 (UTC)
degree of a function being "D", can |D|<1, while squreroots are 1/2 i'm not very sure, on this. but it isn't adressed in the artical
This is completely bogus. The degree of f(x) = 0 is 0. Ceroklis 20:17, 10 April 2007 (UTC)
I does make sense, I have now figured it out. I guess the proper way to define this would go along those lines:
A
polynomial over a
ring is a function such that is finite.
The degree of a polynomial is the highest n such that . Obviously this would then not be defined for the zero polynomial.
We should have these formal definitions somewhere, and then everything follows, that R[X] is a ring, that deg is a valuation, etc... Right now there are bits and pieces spread among the
polynomial,
degree of a polynomial and
polynomial ring articles. Any idea on how to unify these articles ?
grubber: I have noted your new section on abstract algebra. This is a good idea but as said above it should be based on formal definitions and united with the other articles. In particular, the last sentence is misleading. deg(0) is not undefined because the norm is undefined at zero, it is undefined due to the way deg was defined. Deriving stuff in the proper order is important.
Ceroklis
23:19, 11 April 2007 (UTC)
i is the squart root of -1. So can someone explain what degree i and its factors are? -- pizza1512 Talk Autograph 20:02, 13 April 2007 (UTC)
74.244.68.148 ( talk) 17:04, 15 March 2010 (UTC)
Why is hectic preferred over centic?? Georgia guy ( talk) 00:01, 4 March 2011 (UTC)
The progress of the theory of stability of various types of functional equations such as quadratic, cubic, quartic, quintic, sextic, septic, octic, nonic, decic, undecic, duodecic, tredecic, quattordecic have been dealt by many mathematicians and there are lot of interesting and significant results available in the literature.They go up to 20 in Table 1 of this arXiv preprint. That said, they seem to get quite rare after 10, certainly after 12. Double sharp ( talk) 11:47, 15 November 2022 (UTC)
Hectic links to itself - the link should be removed or a stub should be made! Sobeita ( talk) 20:50, 28 October 2011 (UTC)
"To determine the degree of a polynomial that is not in standard form (for example (y − 3)(2y + 6)( − 4y − 21)) it is easier to expand or express the polynomial into a sum or difference of terms;"
The example given in parentheses contradicts the statement made. If you're in an integral domain in the example, you can take the highest order term in each factor, and add them up. In terms of total number of operations, this is much "easier" than multiplying it all out first. For more general rings, you have to look for zero divisors and so on, but it's not obvious whether or not there is a more efficient method for doing this than doing the entire expansion. In any case, unless there is a definite computational idea in mind, "it is easier" sounds more like an opinion than a fact. — Preceding unsigned comment added by 66.183.55.146 ( talk) 03:51, 7 January 2012 (UTC)
I have corrected the statement and changed the example. D.Lazard ( talk) 11:12, 20 October 2012 (UTC)
Spencerleet has removed two times from the article that -1 is a common convention for the degree of 0. His argument is that he finds this convention not convenient. I agree with him that this convention breaks the formula for the degree of a product. However, the problem is not there. The article asserts (asserted) that some authors use this convention. Undoubtedly, the choice of -1 for the degree of zero is very common, when programming Euclidean division of polynomials, Euclid's algorithm for polynomials, etc., because using -∞ in a program is difficult. For this reason, I'll revert again Spencerleet's edit until he provides, if any, a reliable source asserting that this convention is not common. D.Lazard ( talk) 21:30, 7 July 2014 (UTC)
This edit changed the link for "term" from Term (mathematics) to Addition#Notation. Term (mathematics) was merged into Term (logic) and changed to a redirect. The section Term_(logic)#Elementary_mathematics discusses "term" in the context of polynomials. -- 50.53.53.229 ( talk) 16:29, 18 September 2014 (UTC)
This edit added the phrase "associated rule" three times. The section gives two rules for computing with , but those rules do not use inequalities. -- 50.53.53.229 ( talk) 16:52, 18 September 2014 (UTC)
The lead does not explicitly say that the degree of a (non-zero) polynomial is a non-negative integer. -- 50.53.53.71 ( talk) 18:10, 21 September 2014 (UTC)
Why is x²+xy+y² called a binomial based on having two variables and not a trinomial based on have three terms (in two variables). If you look at the article Trinomial the third example given is 3ts+9t+5s -- which has two variables and is not called a binomial. 192.96.44.56 ( talk) 13:39, 13 March 2024 (UTC)
![]() | This article is rated Start-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||
|
I'm new to this Wikipedia thing, so I'll just post my opinion here before editing. Hope that someone with more experience could help.
The point is that there is no formal (mathematically correct) definition given. What you have is just an explanation/illustration.
Generally a polynomial is defined as a sequence of elements (called coefficients) from a ring indexed by natural numbers with the following characteristic: there is a natural number n (called the degree) so that the n-th coefficient is non-zero and all coefficients with index higher than are zero.
i.e. it is something like P={a0,a1,...,an,0,0,...} where a0,...an are ellements of R and 0 is the zero in R.
More formally: a polynomial is a function P from N (natural numbers) to R (where R is a ring) where there is n in N (the degree) so that P(n) is not zero and for all m>n P(m)=0.
The thing is that the definition of the degree is embedded in the definition of the polynomial. What would be the best way to include here the formal/mathematically correct definition of the degree - should we repeat the polynomial definition here? - AdamSmithee 22:25, 17 November 2005 (UTC)
Two conflicting definitions seem to be given. The first paragraph says the degree is the sum of the powers of all terms, the second says it is the sum of powers in one term. That is confusing. Jewels Vern ( talk) 05:23, 18 October 2013 (UTC)
I've had a bit of a debate with someone regarding degrees 0 and 1 of a polynomial. My argument is this: the degree of is 1, as the exponent on the variable is 1. But the degree of any constant term is 0. E.g: , as 0 is the exponent on x. To my surprise, they were arguing that , therefore the degree is 1. Surely the coefficients can't be used in determining the degree, otherwise the degree of would be 3?! I'm not a mathematician so I'm a bit reluctant to change the article, but would it be reasonable to clarify this in the article? Or maybe I was wrong? -- 146.227.11.232 15:55, 13 January 2006 (UTC)
degree of a function being "D", can |D|<1, while squreroots are 1/2 i'm not very sure, on this. but it isn't adressed in the artical
This is completely bogus. The degree of f(x) = 0 is 0. Ceroklis 20:17, 10 April 2007 (UTC)
I does make sense, I have now figured it out. I guess the proper way to define this would go along those lines:
A
polynomial over a
ring is a function such that is finite.
The degree of a polynomial is the highest n such that . Obviously this would then not be defined for the zero polynomial.
We should have these formal definitions somewhere, and then everything follows, that R[X] is a ring, that deg is a valuation, etc... Right now there are bits and pieces spread among the
polynomial,
degree of a polynomial and
polynomial ring articles. Any idea on how to unify these articles ?
grubber: I have noted your new section on abstract algebra. This is a good idea but as said above it should be based on formal definitions and united with the other articles. In particular, the last sentence is misleading. deg(0) is not undefined because the norm is undefined at zero, it is undefined due to the way deg was defined. Deriving stuff in the proper order is important.
Ceroklis
23:19, 11 April 2007 (UTC)
i is the squart root of -1. So can someone explain what degree i and its factors are? -- pizza1512 Talk Autograph 20:02, 13 April 2007 (UTC)
74.244.68.148 ( talk) 17:04, 15 March 2010 (UTC)
Why is hectic preferred over centic?? Georgia guy ( talk) 00:01, 4 March 2011 (UTC)
The progress of the theory of stability of various types of functional equations such as quadratic, cubic, quartic, quintic, sextic, septic, octic, nonic, decic, undecic, duodecic, tredecic, quattordecic have been dealt by many mathematicians and there are lot of interesting and significant results available in the literature.They go up to 20 in Table 1 of this arXiv preprint. That said, they seem to get quite rare after 10, certainly after 12. Double sharp ( talk) 11:47, 15 November 2022 (UTC)
Hectic links to itself - the link should be removed or a stub should be made! Sobeita ( talk) 20:50, 28 October 2011 (UTC)
"To determine the degree of a polynomial that is not in standard form (for example (y − 3)(2y + 6)( − 4y − 21)) it is easier to expand or express the polynomial into a sum or difference of terms;"
The example given in parentheses contradicts the statement made. If you're in an integral domain in the example, you can take the highest order term in each factor, and add them up. In terms of total number of operations, this is much "easier" than multiplying it all out first. For more general rings, you have to look for zero divisors and so on, but it's not obvious whether or not there is a more efficient method for doing this than doing the entire expansion. In any case, unless there is a definite computational idea in mind, "it is easier" sounds more like an opinion than a fact. — Preceding unsigned comment added by 66.183.55.146 ( talk) 03:51, 7 January 2012 (UTC)
I have corrected the statement and changed the example. D.Lazard ( talk) 11:12, 20 October 2012 (UTC)
Spencerleet has removed two times from the article that -1 is a common convention for the degree of 0. His argument is that he finds this convention not convenient. I agree with him that this convention breaks the formula for the degree of a product. However, the problem is not there. The article asserts (asserted) that some authors use this convention. Undoubtedly, the choice of -1 for the degree of zero is very common, when programming Euclidean division of polynomials, Euclid's algorithm for polynomials, etc., because using -∞ in a program is difficult. For this reason, I'll revert again Spencerleet's edit until he provides, if any, a reliable source asserting that this convention is not common. D.Lazard ( talk) 21:30, 7 July 2014 (UTC)
This edit changed the link for "term" from Term (mathematics) to Addition#Notation. Term (mathematics) was merged into Term (logic) and changed to a redirect. The section Term_(logic)#Elementary_mathematics discusses "term" in the context of polynomials. -- 50.53.53.229 ( talk) 16:29, 18 September 2014 (UTC)
This edit added the phrase "associated rule" three times. The section gives two rules for computing with , but those rules do not use inequalities. -- 50.53.53.229 ( talk) 16:52, 18 September 2014 (UTC)
The lead does not explicitly say that the degree of a (non-zero) polynomial is a non-negative integer. -- 50.53.53.71 ( talk) 18:10, 21 September 2014 (UTC)
Why is x²+xy+y² called a binomial based on having two variables and not a trinomial based on have three terms (in two variables). If you look at the article Trinomial the third example given is 3ts+9t+5s -- which has two variables and is not called a binomial. 192.96.44.56 ( talk) 13:39, 13 March 2024 (UTC)