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  Polynomials and numerical semigroups
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<p><a id="X7D2C77607815273E" name="X7D2C77607815273E"></a></p>
<div class="ChapSects"><a href="chap10_mj.html#X7D2C77607815273E">10 <span class="Heading">
  Polynomials and numerical semigroups
 </span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10_mj.html#X808FAEE28572191C">10.1 <span class="Heading">
  Generating functions or Hilbert series
 </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X8391C8E782FBFA8A">10.1-1 NumericalSemigroupPolynomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X7F59E1167C1EE578">10.1-2 IsNumericalSemigroupPolynomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X855497F77D13436F">10.1-3 NumericalSemigroupFromNumericalSemigroupPolynomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X780479F978D166B0">10.1-4 HilbertSeriesOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X87C88E5C7B56931F">10.1-5 GraeffePolynomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X87A46B53815B158F">10.1-6 IsCyclotomicPolynomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X7D9618ED83776B0B">10.1-7 IsKroneckerPolynomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X8366BB727C496D31">10.1-8 IsCyclotomicNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X7B428FA2877EC733">10.1-9 CyclotomicExponentSequence</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X7A33BA9B813A4070">10.1-10 WittCoefficients</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X82C6355287C3BDD1">10.1-11 IsSelfReciprocalUnivariatePolynomial</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10_mj.html#X7EEF2A1781432A2D">10.2 <span class="Heading">
  Semigroup of values of algebraic curves
 </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X7FFF949A7BEEA912">10.2-1 SemigroupOfValuesOfPlaneCurveWithSinglePlaceAtInfinity</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X834D6B1A7C421B9F">10.2-2 IsDeltaSequence</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X824ABFD680A34495">10.2-3 DeltaSequencesWithFrobeniusNumber</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X87B819B886CA5A5C">10.2-4 CurveAssociatedToDeltaSequence</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X7E2C3E9A7DE7A078">10.2-5 SemigroupOfValuesOfPlaneCurve</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X7F88774F7812D30E">10.2-6 SemigroupOfValuesOfCurve_Local</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X8597259279D1E793">10.2-7 SemigroupOfValuesOfCurve_Global</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X7EE8528484642CEE">10.2-8 GeneratorsModule_Global</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X836D31F787641C22">10.2-9 GeneratorsKahlerDifferentials</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X7A04B8887F493733">10.2-10 IsMonomialNumericalSemigroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10_mj.html#X84C670E1826F8B92">10.3 <span class="Heading">
  Semigroups and Legendrian curves
 </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X7980A7CE79F09A89">10.3-1 LegendrianGenericNumericalSemigroup</a></span>
</div></div>
</div>

<h3>10 <span class="Heading">
  Polynomials and numerical semigroups
 </span></h3>

<p>Polynomials appear related to numerical semigroups in several ways. One of them is through their associated generating function (or Hilbert series), and another via value semigroups of a curve; and curves might be defined by polynomials. In this chapter we present several functions to compute the polynomial and Hilbert series associated to a numerical semigroup, and to calculate the respective numerical semigroups given a set of defining polynomials.</p>

<p><a id="X808FAEE28572191C" name="X808FAEE28572191C"></a></p>

<h4>10.1 <span class="Heading">
  Generating functions or Hilbert series
 </span></h4>

<p>Let <span class="SimpleMath">\(S\)</span> be a numerical semigroup. The Hilbert series or generating function associated to <span class="SimpleMath">\(S\)</span> is <span class="SimpleMath">\(H_S(x)=\sum_{s\in S}x^s\)</span> (actually it is the Hilbert function of the ring <span class="SimpleMath">\(K[S]\)</span> with <span class="SimpleMath">\(K\)</span> a field). See for instance <a href="chapBib_mj.html#biBM">[Mor14]</a>.</p>

<p><a id="X8391C8E782FBFA8A" name="X8391C8E782FBFA8A"></a></p>

<h5>10.1-1 NumericalSemigroupPolynomial</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NumericalSemigroupPolynomial</code>( <var class="Arg">s</var>, <var class="Arg">x</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">s</var> is a numerical semigroups and <var class="Arg">x</var> a variable (or a value to evaluate in). The output is the polynomial <span class="SimpleMath">\(1+(x-1)\sum_{s\in \mathbb N\setminus S} x^s\)</span>, which equals <span class="SimpleMath">\((1-x)H_S(x)\)</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">x:=X(Rationals,"x");;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(5,7,9);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">NumericalSemigroupPolynomial(s,x);</span>
x^14-x^13+x^12-x^11+x^9-x^8+x^7-x^6+x^5-x+1
</pre></div>

<p><a id="X7F59E1167C1EE578" name="X7F59E1167C1EE578"></a></p>

<h5>10.1-2 IsNumericalSemigroupPolynomial</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsNumericalSemigroupPolynomial</code>( <var class="Arg">f</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">f</var> is a polynomial in one variable. The output is true if there exists a numerical semigroup <span class="SimpleMath">\(S\)</span> such that <var class="Arg">f</var> equals <span class="SimpleMath">\((1-x)H_S(x)\)</span>, that is, the polynomial associated to <span class="SimpleMath">\(S\)</span> (false otherwise).</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">x:=X(Rationals,"x");;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(5,6,7,8);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">f:=NumericalSemigroupPolynomial(s,x);</span>
x^10-x^9+x^5-x+1
<span class="GAPprompt">gap></span> <span class="GAPinput">IsNumericalSemigroupPolynomial(f);</span>
true
</pre></div>

<p><a id="X855497F77D13436F" name="X855497F77D13436F"></a></p>

<h5>10.1-3 NumericalSemigroupFromNumericalSemigroupPolynomial</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NumericalSemigroupFromNumericalSemigroupPolynomial</code>( <var class="Arg">f</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">f</var> is a polynomial associated to a numerical semigroup (otherwise yields error). The output is the numerical semigroup <span class="SimpleMath">\(S\)</span> such that <var class="Arg">f</var> equals <span class="SimpleMath">\((1-x)H_S(x)\)</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">x:=X(Rationals,"x");;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(5,6,7,8);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">f:=NumericalSemigroupPolynomial(s,x);</span>
x^10-x^9+x^5-x+1
<span class="GAPprompt">gap></span> <span class="GAPinput">NumericalSemigroupFromNumericalSemigroupPolynomial(f)=s;</span>
true
</pre></div>

<p><a id="X780479F978D166B0" name="X780479F978D166B0"></a></p>

<h5>10.1-4 HilbertSeriesOfNumericalSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HilbertSeriesOfNumericalSemigroup</code>( <var class="Arg">s</var>, <var class="Arg">x</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">s</var> is a numerical semigroup and <var class="Arg">x</var> a variable (or a value to evaluate in). The output is the series <span class="SimpleMath">\(\sum_{s\in S} x^s\)</span>. The series is given as a rational function.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">x:=X(Rationals,"x");;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(5,7,9);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">HilbertSeriesOfNumericalSemigroup(s,x);</span>
(x^14-x^13+x^12-x^11+x^9-x^8+x^7-x^6+x^5-x+1)/(-x+1)
</pre></div>

<p><a id="X87C88E5C7B56931F" name="X87C88E5C7B56931F"></a></p>

<h5>10.1-5 GraeffePolynomial</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GraeffePolynomial</code>( <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">p</var> is a polynomial. Computes the Graeffe polynomial of <var class="Arg">p</var>. Needed to test if <var class="Arg">p</var> is a cyclotomic polynomial (see <a href="chapBib_mj.html#biBBD-cyclotomic">[BD89]</a>).</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">x:=Indeterminate(Rationals,1);; SetName(x,"x");</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">GraeffePolynomial(x^2-1);</span>
x^2-2*x+1
</pre></div>

<p><a id="X87A46B53815B158F" name="X87A46B53815B158F"></a></p>

<h5>10.1-6 IsCyclotomicPolynomial</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCyclotomicPolynomial</code>( <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">p</var> is a polynomial. Detects if <var class="Arg">p</var> is a cyclotomic polynomial using the procedure given in <a href="chapBib_mj.html#biBBD-cyclotomic">[BD89]</a>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">CyclotomicPolynomial(Rationals,3);</span>
x^2+x+1
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCyclotomicPolynomial(last);</span>
true
</pre></div>

<p><a id="X7D9618ED83776B0B" name="X7D9618ED83776B0B"></a></p>

<h5>10.1-7 IsKroneckerPolynomial</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsKroneckerPolynomial</code>( <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">p</var> is a polynomial. Detects if <var class="Arg">p</var> is a Kronecker polynomial, that is, a monic polynomial with integer coefficients having all its roots in the unit circumference, or equivalently, a product of cyclotomic polynomials. The current implementation has been done with A. Herrera-Poyatos, following <a href="chapBib_mj.html#biBBD-cyclotomic">[BD89]</a>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">x:=X(Rationals,"x");;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput"> s:=NumericalSemigroup(3,5,7);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput"> t:=NumericalSemigroup(4,6,9);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">p:=NumericalSemigroupPolynomial(s,x);</span>
x^5-x^4+x^3-x+1
<span class="GAPprompt">gap></span> <span class="GAPinput">q:=NumericalSemigroupPolynomial(t,x);</span>
x^12-x^11+x^8-x^7+x^6-x^5+x^4-x+1
<span class="GAPprompt">gap></span> <span class="GAPinput">IsKroneckerPolynomial(p);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsKroneckerPolynomial(q);</span>
true
</pre></div>

<p><a id="X8366BB727C496D31" name="X8366BB727C496D31"></a></p>

<h5>10.1-8 IsCyclotomicNumericalSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCyclotomicNumericalSemigroup</code>( <var class="Arg">s</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">s</var> is a numerical semigroup. Detects if the polynomial associated to <var class="Arg">s</var> is a Kronecker polynomial.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">l:=CompleteIntersectionNumericalSemigroupsWithFrobeniusNumber(21);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ForAll(l,IsCyclotomicNumericalSemigroup);</span>
true
</pre></div>

<p><a id="X7B428FA2877EC733" name="X7B428FA2877EC733"></a></p>

<h5>10.1-9 CyclotomicExponentSequence</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CyclotomicExponentSequence</code>( <var class="Arg">s</var>, <var class="Arg">k</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><var class="Arg">s</var> is a numerical semigroup and <var class="Arg">k</var> is a positive integer. Outputs the list of the first <var class="Arg">k</var> elements of the cyclotomic exponent sequence of s (see <a href="chapBib_mj.html#biBC-GS-M">[CGM16]</a>).</p>

<p>The sequence will be truncated if the semigroup is cyclotomic and k is bigger than the last nonzero element in its sequence.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(3,4);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">CyclotomicExponentSequence(s,20);</span>
[ 1, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(3,5,7);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">CyclotomicExponentSequence(s,20);</span>
[ 1, 0, -1, 0, -1, 0, -1, 0, 0, 1, 0, 1, 0, 1, 0, 0, -1, 0, -1, 0 ]
</pre></div>

<p><a id="X7A33BA9B813A4070" name="X7A33BA9B813A4070"></a></p>

<h5>10.1-10 WittCoefficients</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ WittCoefficients</code>( <var class="Arg">p</var>, <var class="Arg">k</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><var class="Arg">p</var> is a univariate polynomial with integer coefficientas and <span class="SimpleMath">\(p(1)=1\)</span>. Then <span class="SimpleMath">\(p(x)=\prod_{n\ge 0}(1-x^n)^{e_n}\)</span>, for some integers <span class="SimpleMath">\(e_n\)</span>. The output is the list <span class="SimpleMath">\([e_1,\ldots,e_k]\)</span>, and it is computed by using <a href="chapBib_mj.html#biBC-GS-HP-M">[CGHM19]</a>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(3,4);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">x:=Indeterminate(Rationals,"x");;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">p:=NumericalSemigroupPolynomial(s,x);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">WittCoefficients(p,20);</span>
[ 1, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ]
</pre></div>

<p>The difference with this example and the one in <code class="func">CyclotomicExponentSequence</code> (<a href="chap10_mj.html#X7B428FA2877EC733"><span class="RefLink">10.1-9</span></a>) is that in that case a cyclotomic check is performed that slows down the process.</p>

<p><a id="X82C6355287C3BDD1" name="X82C6355287C3BDD1"></a></p>

<h5>10.1-11 IsSelfReciprocalUnivariatePolynomial</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSelfReciprocalUnivariatePolynomial</code>( <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">p</var> is a univariate polynomial. Detects if <var class="Arg">p</var> is selfreciprocal. A numerical semigroup is symmetric if and only if it is selfreciprocal, <a href="chapBib_mj.html#biBM">[Mor14]</a>. The current implementation is due to A. Herrera-Poyatos.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">l:=IrreducibleNumericalSemigroupsWithFrobeniusNumber(13);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">x:=X(Rationals,"x");;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ForAll(l, s-></span>
<span class="GAPprompt">></span> <span class="GAPinput">IsSelfReciprocalUnivariatePolynomial(NumericalSemigroupPolynomial(s,x)));</span>
true
</pre></div>

<p><a id="X7EEF2A1781432A2D" name="X7EEF2A1781432A2D"></a></p>

<h4>10.2 <span class="Heading">
  Semigroup of values of algebraic curves
 </span></h4>

<p>Let <span class="SimpleMath">\(f(x,y)\in \mathbb K[x,y]\)</span>, with <span class="SimpleMath">\(\mathbb K\)</span> an algebraically closed field of characteristic zero. Let <span class="SimpleMath">\(f(x,y)=y^n+a_1(x)y^{n-1}+\dots+a_n(x)\)</span> be a nonzero polynomial of <span class="SimpleMath">\(\mathbb K[x][y]\)</span>. After possibly a change of variables, we may assume that, that <span class="SimpleMath">\(\deg_x(a_i(x))\le i-1\)</span> for all <span class="SimpleMath">\(i\in\{1,\ldots, n\}\)</span>. For <span class="SimpleMath">\(g\in\mathbb K[x,y]\)</span> that is not a multiple of <span class="SimpleMath">\(f\)</span>, define <span class="SimpleMath">\(\mathrm{int}(f,g)=\dim_\mathbb K \frac{\mathbb K[x,y]}{(f,g)}\)</span>. If <span class="SimpleMath">\(f\)</span> has one place at infinity, then the set <span class="SimpleMath">\(\{\mathrm{int}(f,g)\mid g\in\mathbb K[x,y]\setminus(f)\}\)</span> is a free numerical semigroup (and thus a complete intersection).</p>

<p><a id="X7FFF949A7BEEA912" name="X7FFF949A7BEEA912"></a></p>

<h5>10.2-1 SemigroupOfValuesOfPlaneCurveWithSinglePlaceAtInfinity</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SemigroupOfValuesOfPlaneCurveWithSinglePlaceAtInfinity</code>( <var class="Arg">f</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">f</var> is a polynomial in the variables X(Rationals,1) and X(Rationals,2). Computes the semigroup <span class="SimpleMath">\(\{\mathrm{int}(f,g)\mid g\in\mathbb K[x,y]\setminus(f)\}\)</span>, where <span class="SimpleMath">\(\mathrm{int}(f,g)=\dim_{\mathbb K} (\mathbb K[x,y]/(f,g))\)</span>. The algorithm checks if <var class="Arg">f</var> has one place at infinity. If the extra argument "all" is given, then the output is the <span class="SimpleMath">\(\delta\)</span>-sequence and approximate roots of <var class="Arg">f</var>. The method is explained in <a href="chapBib_mj.html#biBAGS14">[AG16a]</a>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">x:=Indeterminate(Rationals,1);; SetName(x,"x");</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">y:=Indeterminate(Rationals,2);; SetName(y,"y");</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">f:=((y^3-x^2)^2-x*y^2)^4-(y^3-x^2);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SemigroupOfValuesOfPlaneCurveWithSinglePlaceAtInfinity(f,"all");</span>
[ [ 24, 16, 28, 7 ], [ y, y^3-x^2, y^6-2*x^2*y^3+x^4-x*y^2 ] ]
</pre></div>

<p><a id="X834D6B1A7C421B9F" name="X834D6B1A7C421B9F"></a></p>

<h5>10.2-2 IsDeltaSequence</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsDeltaSequence</code>( <var class="Arg">l</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">l</var> is a list of positive integers. Assume that <var class="Arg">l</var> equals <span class="SimpleMath">\(a_0,a_1,\dots,a_h\)</span>. Then <var class="Arg">l</var> is a <span class="SimpleMath">\(\delta\)</span>-sequence if <span class="SimpleMath">\(\gcd(a_0,\ldots, a_h)=1\)</span>, <span class="SimpleMath">\(\langle a_0,\cdots, a_s\rangle\)</span> is free, <span class="SimpleMath">\(a_kD_k > a_{k+1}D_{k+1}\)</span> and <span class="SimpleMath">\(a_0> a_1 > D_2 > D_3 > \ldots > D_{h+1}\)</span>, where <span class="SimpleMath">\(D_1=a_0\)</span>, <span class="SimpleMath">\(D_k=\gcd(D_{k-1},a_{k-1})\)</span>.</p>

<p>Every <span class="SimpleMath">\(\delta\)</span>-sequence generates a numerical semigroup that is the semigroup of values of a plane curve with one place at infinity.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsDeltaSequence([24,16,28,7]);</span>
true
</pre></div>

<p><a id="X824ABFD680A34495" name="X824ABFD680A34495"></a></p>

<h5>10.2-3 DeltaSequencesWithFrobeniusNumber</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DeltaSequencesWithFrobeniusNumber</code>( <var class="Arg">f</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">f</var> is an integer. Computes the set of all <span class="SimpleMath">\(\delta\)</span>-sequences generating numerical semigroups with Frobenius number <var class="Arg">f</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">DeltaSequencesWithFrobeniusNumber(21);</span>
[ [ 8, 6, 11 ], [ 10, 4, 15 ], [ 12, 8, 6, 11 ], [ 14, 4, 11 ],
  [ 15, 10, 4 ], [ 23, 2 ] ]
</pre></div>

<p><a id="X87B819B886CA5A5C" name="X87B819B886CA5A5C"></a></p>

<h5>10.2-4 CurveAssociatedToDeltaSequence</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CurveAssociatedToDeltaSequence</code>( <var class="Arg">l</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">l</var> is a <span class="SimpleMath">\(\delta\)</span>-sequence. Computes a curve in the variables X(Rationals,1) and X(Rationals,2) whose semigroup of values is generated by the <var class="Arg">l</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">CurveAssociatedToDeltaSequence([24,16,28,7]);</span>
y^24-8*x^2*y^21+28*x^4*y^18-56*x^6*y^15-4*x*y^20+70*x^8*y^12+24*x^3*y^17-56*x^\
10*y^9-60*x^5*y^14+28*x^12*y^6+80*x^7*y^11+6*x^2*y^16-8*x^14*y^3-60*x^9*y^8-24\
*x^4*y^13+x^16+24*x^11*y^5+36*x^6*y^10-4*x^13*y^2-24*x^8*y^7-4*x^3*y^12+6*x^10\
*y^4+8*x^5*y^9-4*x^7*y^6+x^4*y^8-y^3+x^2
<span class="GAPprompt">gap></span> <span class="GAPinput">SemigroupOfValuesOfPlaneCurveWithSinglePlaceAtInfinity(last,"all");</span>
[ [ 24, 16, 28, 7 ], [ y, y^3-x^2, y^6-2*x^2*y^3+x^4-x*y^2 ] ]
</pre></div>

<p><a id="X7E2C3E9A7DE7A078" name="X7E2C3E9A7DE7A078"></a></p>

<h5>10.2-5 SemigroupOfValuesOfPlaneCurve</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SemigroupOfValuesOfPlaneCurve</code>( <var class="Arg">f</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">f</var> is a polynomial in the variables X(Rationals,1) and X(Rationals,2). The singular package is mandatory. Either by loading it prior to numerical semigroups or by using <var class="Arg">NumSgpsUseSingular()</var>. If <span class="SimpleMath">\(f\)</span> is irreducible, computes the semigroup <span class="SimpleMath">\(\{\mathrm{int}(f,g)\mid g\in\mathbb K[x,y]\setminus(f)\}\)</span>, where <span class="SimpleMath">\(\mathrm{int}(f,g)=\dim_{\mathbb K} (\mathbb K[\![x,y]\!]/(f,g))\)</span>. If it has two components, the output is the value semigroup in two variables, and thus a good semigroup. If there are more components, then the output is that of <var class="Arg">semigroup</var> in the alexpoly singular library.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">x:=X(Rationals,"x");; y:=X(Rationals,"y");;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">f:= y^4-2*x^3*y^2-4*x^5*y+x^6-x^7;</span>
-x^7+x^6-4*x^5*y-2*x^3*y^2+y^4
<span class="GAPprompt">gap></span> <span class="GAPinput">SemigroupOfValuesOfPlaneCurve(f);</span>
<Numerical semigroup with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimalGenerators(last);</span>
[ 4, 6, 13 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">f:=(y^4-2*x^3*y^2-4*x^5*y+x^6-x^7)*(y^2-x^3);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SemigroupOfValuesOfPlaneCurve(f);</span>
<Good semigroup>
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimalGenerators(last);</span>
[ [ 4, 2 ], [ 6, 3 ], [ 13, 15 ], [ 29, 13 ] ]
</pre></div>

<p><a id="X7F88774F7812D30E" name="X7F88774F7812D30E"></a></p>

<h5>10.2-6 SemigroupOfValuesOfCurve_Local</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SemigroupOfValuesOfCurve_Local</code>( <var class="Arg">arg</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>The function admits one or two parameters. In any case, the first is a list of polynomials <var class="Arg">pols</var>. And the second can be the string <var class="Arg">"basis"</var> or an integer <var class="Arg">val</var>.</p>

<p>If only one argument is given, the output is the semigroup of all possible orders of <span class="SimpleMath">\(K[[pols]]\)</span> provided that <span class="SimpleMath">\(K[[x]]/K[[pols]]\)</span> has finite length. If the second argument <var class="Arg">"basis"</var> is given, then the output is a (reduced) basis of the algebra <span class="SimpleMath">\(K[[pols]]\)</span> such that the orders of the basis elements generate minimally the semigroup of orders of <span class="SimpleMath">\(K[[pols]]\)</span>. If an integer <var class="Arg">val</var> is the second argument, then the output is a polynomial in <span class="SimpleMath">\(K[[pols]]\)</span> with order <var class="Arg">val</var> (<code class="code">fail</code> if there is no such polynomial, that is, <var class="Arg">val</var> is not in the semigroup of values).</p>

<p>The method is explained in <a href="chapBib_mj.html#biBAGSM14">[AGM17]</a>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">x:=Indeterminate(Rationals,"x");;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SemigroupOfValuesOfCurve_Local([x^4,x^6+x^7,x^13]);</span>
<Numerical semigroup with 4 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimalGeneratingSystem(last);</span>
[ 4, 6, 13, 15 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">SemigroupOfValuesOfCurve_Local([x^4,x^6+x^7,x^13], "basis");</span>
[ x^4, x^7+x^6, x^13, x^15 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">SemigroupOfValuesOfCurve_Local([x^4,x^6+x^7,x^13], 20);</span>
x^20
</pre></div>

<p><a id="X8597259279D1E793" name="X8597259279D1E793"></a></p>

<h5>10.2-7 SemigroupOfValuesOfCurve_Global</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SemigroupOfValuesOfCurve_Global</code>( <var class="Arg">arg</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>The function admits one or two parameters. In any case, the first is a list of polynomials <var class="Arg">pols</var>. And the second can be the string <var class="Arg">"basis"</var> or an integer <var class="Arg">val</var>.</p>

<p>If only one argument is given, the output is the semigroup of all possible degrees of <span class="SimpleMath">\(K[pols]\)</span> provided that <span class="SimpleMath">\(K[x]/K[pols]\)</span> has finite length. If the second argument <var class="Arg">"basis"</var> is given, then the output is a (reduced) basis of the algebra <span class="SimpleMath">\(K[pols]\)</span> such that the degrees of the basis elements generate minimally the semigroup of degrees of <span class="SimpleMath">\(K[pols]\)</span>. If an integer <var class="Arg">val</var> is the second argument, then the output is a polynomial in <span class="SimpleMath">\(K[pols]\)</span> with degree <var class="Arg">val</var> (<code class="code">fail</code> if there is no such polynomial, that is, <var class="Arg">val</var> is not in the semigroup of values).</p>

<p>The method is explained in <a href="chapBib_mj.html#biBAGSM14">[AGM17]</a>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">x:=Indeterminate(Rationals,"x");;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SemigroupOfValuesOfCurve_Global([x^4,x^6+x^7,x^13]);</span>
<Numerical semigroup with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimalGeneratingSystem(last);</span>
[ 4, 7, 13 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">SemigroupOfValuesOfCurve_Global([x^4,x^6+x^7,x^13],"basis");</span>
[ x^4, x^7+x^6, x^13 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">SemigroupOfValuesOfCurve_Global([x^4,x^6+x^7,x^13],12);</span>
x^12
<span class="GAPprompt">gap></span> <span class="GAPinput">SemigroupOfValuesOfCurve_Global([x^4,x^6+x^7,x^13],6);</span>
fail
</pre></div>

<p><a id="X7EE8528484642CEE" name="X7EE8528484642CEE"></a></p>

<h5>10.2-8 GeneratorsModule_Global</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneratorsModule_Global</code>( <var class="Arg">A</var>, <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">A</var> and <var class="Arg">M</var> are lists of polynomials in the same variable. The output is a basis of the ideal <span class="SimpleMath">\(M K[A]\)</span>, that is, a set <span class="SimpleMath">\(F\)</span> such that <span class="SimpleMath">\(deg(F)\)</span> generates the ideal <span class="SimpleMath">\(deg(M K[A])\)</span> of <span class="SimpleMath">\(deg(K[A])\)</span>, where <span class="SimpleMath">\(deg\)</span> stands for degree. The method is explained in <a href="chapBib_mj.html#biBA-A-GS">[AAG19]</a>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">t:=Indeterminate(Rationals,"t");;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:=[t^6+t,t^4];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M:=[t^3,t^4];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsModule_Global(A,M);</span>
[ t^3, t^4, t^5, t^6 ]
</pre></div>

<p><a id="X836D31F787641C22" name="X836D31F787641C22"></a></p>

<h5>10.2-9 GeneratorsKahlerDifferentials</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneratorsKahlerDifferentials</code>( <var class="Arg">A</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">A</var> is a list of polynomials in the same variable. The output is <var class="Arg">GeneratorsModule_Global(A,M)</var>, with <var class="Arg">M</var> the set of derivatives of the elements in <var class="Arg">A</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">t:=Indeterminate(Rationals,"t");;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsKahlerDifferentials([t^3,t^4]);</span>
[ t^2, t^3 ]
</pre></div>

<p><a id="X7A04B8887F493733" name="X7A04B8887F493733"></a></p>

<h5>10.2-10 IsMonomialNumericalSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsMonomialNumericalSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p><var class="Arg">S</var> is a numerical semigroup. Tests whether <var class="Arg">S</var> a monomial numerical semigroup.</p>

<p>Let <span class="SimpleMath">\(R\)</span> a Noetherian ring such that <span class="SimpleMath">\(K \subseteq R \subseteq K[[t]]\)</span>, <span class="SimpleMath">\(K\)</span> is a field of characteristic zero, the algebraic closure of <span class="SimpleMath">\(R\)</span> is <span class="SimpleMath">\(K[[t]]\)</span>, and the conductor <span class="SimpleMath">\((R : K[[t]])\)</span> is not zero. If <span class="SimpleMath">\(v : K((t))\to {\mathbb Z}\)</span> is the natural valuation for <span class="SimpleMath">\(K((t))\)</span>, then <span class="SimpleMath">\(v(R)\)</span> is a numerical semigroup.</p>

<p>Let <span class="SimpleMath">\(S\)</span> be a numerical semigroup minimally generated by <span class="SimpleMath">\(\{n_1,\ldots,n_e\}\)</span>. The semigroup ring associated to <span class="SimpleMath">\(S\)</span> is <span class="SimpleMath">\(K[[S]]=K[[t^{n_1},\ldots,t^{n_e}]]\)</span>. A ring is called a semigroup ring if it is of the form <span class="SimpleMath">\(K[[S]]\)</span>, for some numerical semigroup <span class="SimpleMath">\(S\)</span>. We say that <span class="SimpleMath">\(S\)</span> is a monomial numerical semigroup if for any <span class="SimpleMath">\(R\)</span> as above with <span class="SimpleMath">\(v(R)=S\)</span>, <span class="SimpleMath">\(R\)</span> is a semigroup ring. See <a href="chapBib_mj.html#biBVMic02">[Mic02]</a> for details.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMonomialNumericalSemigroup(NumericalSemigroup(4,6,7));</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMonomialNumericalSemigroup(NumericalSemigroup(4,6,11));</span>
false
</pre></div>

<p><a id="X84C670E1826F8B92" name="X84C670E1826F8B92"></a></p>

<h4>10.3 <span class="Heading">
  Semigroups and Legendrian curves
 </span></h4>

<p><a id="X7980A7CE79F09A89" name="X7980A7CE79F09A89"></a></p>

<h5>10.3-1 LegendrianGenericNumericalSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LegendrianGenericNumericalSemigroup</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">n</var> and <var class="Arg">m</var> are coprime integers with <span class="SimpleMath">\(\textit{m}\ge 2\textit{n}+1\)</span>. The output is the semigroup of a generic element in the class of irreducible Legendrian singularities with equisingularity equal to the topological type of <span class="SimpleMath">\(y^n=x^m\)</span>, as described in Section 5 of <a href="chapBib_mj.html#biBaraujo-neto">[AN09]</a>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=LegendrianGenericNumericalSemigroup(5,11);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SmallElements(s);</span>
[ 0, 5, 6, 10, 11, 12, 13, 15 ]
</pre></div>


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