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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
Posted in Notes
Tagged cryptography, derivatives, factor theorem, polynomials, remainder theorem, ring theory, secret sharing
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