Proof of a Positive and Infinitely Small Polynomial
Date: 10 Apr 1995 09:53:43 -0400 From: Marek Kacprzak Subject: Polynomials Dear Mathematical Society! I have a math problem! If You are able to help me I will be very grateful to You. And the Problem is following: Prove ,that there exists a two variables polynomial W(x,y) such that for any x and y it is always positive but at the same time infinitely small. Please help me to solve it. Marek Kacprzak
Date: 10 Apr 1995 19:20:09 -0400 From: Dr. Ken Subject: Re: Polynomials Hello there! Well, I'm not sure I understand the problem. The problem is that there is no number that is infinitely small and still positive. Perhaps something is lost in translation? If you can clarify the problem any more, please write back. -Ken "Dr." Math
Date: 11 Apr 1995 05:43:08 -0400 From: Marek Kacprzak Subject: Re: Polynomials Dear Dr Williams! Perhaps I have written my problem wrong. I don't know: Does this polynomial really exist? This is the question. And now I give the problem once again: Does there exist a two variables polynomial W(x,y) such that for any x,y it is always positive (everywhere) W(x,y)>0 and at the same time infinitely small.Maby it doesnt exist,but it must be checked by a mathematical way (Proof)
Date: 16 Apr 1995 13:09:21 -0400 From: Dr. Ken Subject: Re: Polynomials Hello there! Well, I don't think that such a polynomial can exist, because what you're asking is that the polynomial has a nonexistent value for all x and y, namely a value that is smaller than all positive numbers but greater than zero; because if a non-negative number is less than all positive numbers, then it's dead zero. You could, however, give a different name for the zero function, which will _look_ like it's a function that you're looking for, but in fact it's just the zero function: F(x,y) = Limit as n goes to Infinity of1/(x^2 + y^2 + 2)^n -Ken "Dr." Math
Date: 21 Apr 1995 22:39:13 -0400 From: Marek Kacprzak Subject: Polynomial Dear Dr Williams! This Polynomial exists-I am 100% sure,that this polynomial exists. I need only a Proof! Please try to help me find a solution! Try to consult with other people who can know something more about this type of problem about this problem. Once again, it is as follows: Prove,that exists a two variable Polynomial W(x,y) ( not function) such that is always positive and at the same time infinitely small.
Date: 21 Apr 1995 22:39:13 -0400 From: Dr. Ken Subject: Re: Polynomial Hello. When you wrote before, you said that you were looking for a proof _or_ a disproof of this phenomenon. How do you know that such a structure exists? Here is some additional material. The size of _any_ polynomial with coefficients in the real numbers will tend toward infinity as its input values tend toward infinity, unless the polynomial is a constant polynomial. In that case, the question becomes "is there a constant number c such that c is positive and infinitely small?" Well, that depends. Some people allow infinitesimal numbers in the real numbers (I personally don't think about them like that, but I suppose it's a personal decision). An infinitesimal number is one that is positive but less than every positive real number. So if you allow polynomials with infinitesimal coefficients, then sure, use a constant polynomial whose value is infinitesimal. But it's still not a very interesting solution. My thanks to Professor Grinstead and Ethan Magness for discussing your question with me. -Ken "Dr." Math
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