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Tag Archives: polynomials
Polynomials and Representations V
It was clear from the earlier articles that n (number of variables ) plays a minimal role in the combinatorics of the symmetric polynomials. Hence, removing the parameter n turns out to be quite convenient; the process gives us the formal ring of symmetric functions. … Continue reading
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Tagged combinatorics, partitions, polynomials, symmetric polynomials
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Polynomials and Representations IV
Power Sum Polynomials The power sum polynomial is defined as follows: In this case, we do not define , although it seems natural to set As before, for a partition define: Note that we must have above since we have … Continue reading
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Tagged partitions, polynomials, representation theory, symmetric polynomials
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Polynomials and Representations III
Complete Symmetric Polynomials Corresponding to the elementary symmetric polynomial, we define the complete symmetric polynomials in to be: For example when , we have: Thus, written as a sum of monomial symmetric polynomials, we have Note that while the elementary symmetric polynomials only go … Continue reading
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Tagged combinatorics, partitions, polynomials, representation theory, symmetric polynomials
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Polynomials and Representations II
More About Partitions Recall that a partition is a sequence of weakly decreasing nonnegative integers, where appending or dropping ending zeros gives us the same partition. A partition is usually represented graphically as a table of boxes or dots: We will … Continue reading
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Tagged combinatorics, partitions, polynomials, representation theory, symmetric polynomials
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Polynomials and Representations I
We have already seen symmetric polynomials and some of their applications in an earlier article. Let us delve into this a little more deeply. Consider the ring of integer polynomials. The symmetric group acts on it by permuting the variables; specifically, … Continue reading
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Tagged combinatorics, partitions, polynomials, representation theory, symmetric polynomials
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Introduction to Ring Theory (7)
Polynomial Rings A polynomial over a ring R is an expression of the form: , where , and . Let’s get some standard terminology out of the way. The element ai is called the coefficient of xi. The largest n for which an ≠ 0 is called … Continue reading
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Tagged cryptography, derivatives, factor theorem, polynomials, remainder theorem, ring theory, secret sharing
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Symmetric Polynomials (III)
Now we generalise this to n variables: . It’s clear what the corresponding building blocks of symmetric polynomials would be: ; ; ; … . We call these ei‘s the elementary symmetric polynomials in the xi‘s. Note that each ei is the coefficient of Ti in the … Continue reading
Symmetric Polynomials (II)
When we move on to n=3 variables, we now have, as basic building blocks, These are just the coefficients of in the expansion of . Once again, any symmetric polynomial in x, y, z with integer coefficients can be expressed as a polynomial … Continue reading
Symmetric Polynomials (I)
[ Background required: knowledge of basic algebra and polynomial operations. ] After a spate of posts on nonIMO related topics, we’re back on track. Here, we shall look at polynomials in n variables, e.g. P(x, y, z) when n = 3. Such … Continue reading