By Bernd Sturmfels

ISBN-10: 3211774165

ISBN-13: 9783211774168

ISBN-10: 3211774173

ISBN-13: 9783211774175

J. Kung and G.-C. Rota, of their 1984 paper, write: “Like the Arabian phoenix emerging out of its ashes, the idea of invariants, suggested lifeless on the flip of the century, is once more on the vanguard of mathematics”. The publication of Sturmfels is either an easy-to-read textbook for invariant concept and a not easy study monograph that introduces a brand new method of the algorithmic facet of invariant idea. The Groebner bases process is the most device through which the imperative difficulties in invariant thought turn into amenable to algorithmic suggestions. scholars will locate the publication a simple advent to this “classical and new” quarter of arithmetic. Researchers in arithmetic, symbolic computation, and computing device technological know-how gets entry to a wealth of study rules, tricks for purposes, outlines and information of algorithms, labored out examples, and learn problems.

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**Extra resources for Algorithms in Invariant Theory**

**Sample text**

The receiver argues that it is more likely that 00000 was sent (and two errors occurred) than that 11111 was sent (and three errors occurred), and therefore decodes 10100 as 00000 D YES. For in some sense 10100 is closer to 00000 than to 11111. v1 ; : : : ; vn / to be the number of places where ui 6D vi . It is easily checked that “dist” is a metric. Then the receiver should decode the received vector as the closest code word, measured in the Hamming distance. Notice that in the above example two errors were corrected.

The following 16 code words 00000000 11101000 01110100 00111010 10011100 01001110 10100110 11010010 11111111 00010111 10001011 11000101 01100011 10110001 01011001 00101101 define a self-dual Œ8; 4; 4 code C2 . Its weight enumerator equals WC2 D x18 C 14x14 x24 C x28 . The following theorem relates the weight enumerators of dual pairs of codes. A proof can be found in Sloane (1977: theorem 6). 8. If C is an Œn; k; d binary code with dual code C ? , then WC ? x1 C x2 ; x1 2k x2 /: 36 Invariant theory of finite groups The class of self-dual codes is of particular interest in coding theory because here the decoding of messages is relatively easy (Sloane, 1977: sect.

8 and the fact that WC is homogeneous of degree n D 2k. The second one is equivalent to the fact that every w 2 C has an even number of 1s, since w w D 0. 9 in the language of invariant theory. Consider the group D8 which is generated by the matrices p12 11 11 and 10 10 . It consists of 16 elements, and geometrically speaking, D8 is the symmetry group of a regular octagon in the plane. 90 . Let WC be the weight enumerator of a self-dual binary code C. Then WC is a polynomial invariant of the group D8 .

### Algorithms in Invariant Theory by Bernd Sturmfels

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