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<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="_normal_spforms.html" title="">normal forms</a></div>
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<div><h1>normal forms</h1>
<div>Let R = k[x<sub>1</sub>, ..., x<sub>n</sub>] be a polynomial ring over a field k, and let <i>I ⊂ R</i> be an ideal. Let <i>{g<sub>1</sub>, ..., g<sub>t</sub>}</i> be a Groebner basis for <i>I</i>. For any <i>f ∈ R</i>, there is a unique ‘remainder’ <i>r ∈ R</i> such that no term of <i>r</i> is divisible by the leading term of any <i>g<sub>i</sub></i> and such that <i>f-r</i> belongs to <i>I</i>. This polynomial <i>r</i> is sometimes called the normal form of <i>f</i>.<p/>
For an example, consider symmetric polynomials. The normal form of the symmetric polynomial <tt>f</tt> with respect to the ideal <tt>I</tt> below writes <tt>f</tt> in terms of the elementary symmetric functions <tt>a,b,c</tt>.<table class="examples"><tr><td><pre>i1 : R = QQ[x,y,z,a,b,c,MonomialOrder=>Eliminate 3];</pre>
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<tr><td><pre>i2 : I = ideal(a-(x+y+z), b-(x*y+x*z+y*z), c-x*y*z)
o2 = ideal (- x - y - z + a, - x*y - x*z - y*z + b, - x*y*z + c)
o2 : Ideal of R</pre>
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<tr><td><pre>i3 : f = x^3+y^3+z^3
3 3 3
o3 = x + y + z
o3 : R</pre>
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<tr><td><pre>i4 : f % I
3
o4 = a - 3a*b + 3c
o4 : R</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="___Gröbner_spbases.html" title="">Gröbner bases</a></span></li>
<li><span><a href="_methods_spfor_spnormal_spforms_spand_spremainder.html" title="calculate the normal form of ring elements and matrices">RingElement % Ideal</a> -- calculate the normal form of ring elements and matrices</span></li>
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