Talk:Degree of a polynomial

message: 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)
replies: text: Please, when you post a comment which does not belongs to an existing thread, put it in a new section at the end of the page. The "New section" button at the top of the talk page allow to do this easily. I so not understand you post: The first paragraph says correctly "The degree of a polynomial is the highest degree of its terms". "Highest" is not the same as "sum". D.Lazard (talk) 09:22, 18 October 2013 (UTC)
thread_title:  Formal definition 

message: 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 10x is 1, as the exponent on the variable is 1. But the degree of any constant term is 0. E.g: 5x^0, as 0 is the exponent on x. To my surprise, they were arguing that 5 = 5^1, therefore the degree is 1. Surely the coefficients can't be used in determining the degree, otherwise the degree of 10x^2 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)
replies: text: The degree = the highest power of X with a non-zero coeff. Indeed, the degree of the polynomial f=5=5X^0 is 0. The factorization of the coeff into primes (if available) doesn't change the degree of the polynomial AdamSmithee 09:07, 16 January 2006 (UTC)
thread_title:  Degrees 0 and 1 

message: 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
:Thanks for asking. See  Degree_of_a_polynomial#The_degree_computed_from_the_function_values. The degree of 3x1/2+5x is 1, while the degree of  3x1/2+17 is 1/2.  I moved your question to the bottom of this discussion page where new contributions are expected to appear. You may sign your discussion entries automatically by four tildes '~'. Bo Jacoby 10:50, 9 August 2006 (UTC)
thread_title:  fractional degree 

message: 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)
replies: text: Over the complex, i is just a non-zero constant, like 2 for example. i.e. i=iz^0+0z^1+0z^2+.... So \deg(i)=0. Furthermore, it is already factored. Ceroklis 21:49, 13 April 2007 (UTC)
thread_title:  Degree of i 

message: 74.244.68.148 (talk) 17:04, 15 March 2010 (UTC)
thread_title:  what is the sum of a polynomials 

message: Why is hectic preferred over centic?? Georgia guy (talk) 00:01, 4 March 2011 (UTC)
replies: text: For that matter, why there is a name for them at all? Even in the rare circumstance of one actually appearing, it would probably just get named "100th degree polynomial". They don't deserve a name any more than (say) 73rd degree ones do (and much less than 11th)... --85.141.139.249 (talk) 20:31, 7 June 2011 (UTC)

text: I think 'hectic' is supposed to be math humor. We'll use 'bajillionic' if we really want, until the 100th degree polynomial actually appears in regular contexts, and then we'll name it appropriately. Sure, not everything gets a name (see 'once', 'twice', 'thrice', 4-ce? quince? ...), but those that do are named humorously at first (see 'velocity', 'acceleration', 'jerk', 'snap', 'crackle', and 'pop'.) Sobeita (talk) 20:50, 28 October 2011 (UTC)
replies: text: The numbers beyond 10 can occasionally be encountered, e.g. here:  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)
thread_title:  Naming a 100th degree polynomial 

message: Hectic links to itself - the link should be removed or a stub should be made! Sobeita (talk) 20:50, 28 October 2011 (UTC)
thread_title:  Hectic's link 

message: "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)
thread_title:  What's the justification for this statement? 

message: 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)
thread_title:  Link to [[term]] 

message: This edit added the phrase "associated rule" three times. The section gives two rules for computing with -\infty, but those rules do not use inequalities. --50.53.53.229 (talk) 16:52, 18 September 2014 (UTC)
replies: text: The wording to which you refer (which I inserted) is not clear and needs to be changed.  What is being referred to is the behaviour of the degree of polynomials from the previous section: that the sum or difference of two polynomials has a degree that is less than or equal to the largest of the degree of each of the polynomials, and the product of two polynomials has a degree that is the sum of the degrees of each of the polynomials. —Quondum 17:30, 18 September 2014 (UTC)
replies: text: Thanks for your clarification. The previous section lists propositions, but there are no citations saying so. The article should refer to propositions and definitions, not behavior and rules. --50.53.53.229 (talk) 18:26, 18 September 2014 (UTC)
replies: text: Actually, deg is a function from non-zero polynomials to non-negative integers, so the section on "Behaviour" might be better called "Properties of the deg function". A problem with the "Behaviour" section is that it doesn't say that the zero polynomial is not in the domain of the deg function. With that in mind, the section on the degree of the zero polynomial can say that the domain of deg can be extended to include the zero polynomial by defining deg 0 = −∞. --50.53.53.229 (talk) 19:11, 18 September 2014 (UTC)
replies: text: That is, by implication, what is already being stated.  But you put it more clearly; cleaning up the presentation in the article would help. —Quondum 20:27, 18 September 2014 (UTC)
thread_title:  What does "associated rule" refer to? 

message: 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)
thread_title:  Degree is a [[non-negative integer]] 

message: 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)
replies: text: x²+xy+y² is nowhere called a binomial. It is a trinomial. "Binary" and "binomial" are two different words that must not be confused. D.Lazard (talk) 15:15, 13 March 2024 (UTC)
replies: text: Excellent! Thank you. I see now I misread and didn't distinguish the two! 192.96.44.56 (talk) 15:30, 13 March 2024 (UTC)
thread_title:  Why is x²+xy+y² called a BI-nomial?