Syzygy (mathematics)

A syzygy is an object from the mathematical field of algebra . If and are two -tuples of polynomials , then syzygy of if is. A syzygy therefore describes how a set of polynomials can be "combined" to form zero. So is z. B. a syzygy of . ${\displaystyle s=(s_{1},\dots ,s_{n})}$ ${\displaystyle g=(g_{1},\dots ,g_{n})}$ ${\displaystyle n}$ ${\displaystyle s}$ ${\displaystyle g}$ ${\displaystyle s_{1}g_{1}+\dots +s_{n}g_{n}=0}$ ${\displaystyle (y,-x)}$ ${\displaystyle (xz,yz)}$

A fundamental theorem about syzygies was proven by David Hilbert in 1890 ( Hilbert's Syzygian Theorem ). Today, syzygies of computer algebra systems are used to manipulate polynomial systems of equations with several unknowns.

Web links

• Eric W. Weisstein : Syzygy . In: MathWorld (English).
• Syzygy . In: Encyclopaedia of Mathematics (English)