Algorithm to find relations between polynomials

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
3
down vote

favorite












Let $p_1,ldots,p_kinmathbbC[x_1,ldots,x_n]$ be polynomials. Is there an algorithm to compute the ideal of relations between them?



More precisely, to find a set of generators for the kernel of the ring homomorphism
$$mathbbC[y_1,ldots,y_k]to mathbbC[x_1,ldots,x_n],quad y_imapsto p_i(x_1,ldots,x_n).$$
(This enables us to compute $mathbbC[p_1,ldots,p_k]$).



Example: Consider
$$x_1x_2,x_3x_4,x_1x_3,x_2x_4inmathbbC[x_1,x_2,x_3,x_4].$$
One obvious relation between them is
$$(x_1x_2)(x_3x_4)=(x_1x_3)(x_2x_4).$$
Thus, if
$$varphi:mathbbC[y_1,y_2,y_3,y_4]tomathbbC[x_1,x_2,x_3,x_4]$$
is the homomorphism defined by
$$y_1mapsto x_1x_2,quad y_2mapsto x_3x_4,quad y_3mapsto x_1x_3,quad y_4mapsto x_2x_4,$$
then $y_1y_2-y_3y_4inker varphi$. It is easy to see that $kervarphi$ is in fact generated by $y_1y_2-y_3y_4$, so
$$mathbbC[x_1x_2,x_3x_4,x_1x_3,x_2x_4]congmathbbC[y_1,y_2,y_3,y_3]/(y_1y_2-y_3y_4).$$



(I posted a similar question in mathematica.stackexchange.)










share|cite|improve this question

























    up vote
    3
    down vote

    favorite












    Let $p_1,ldots,p_kinmathbbC[x_1,ldots,x_n]$ be polynomials. Is there an algorithm to compute the ideal of relations between them?



    More precisely, to find a set of generators for the kernel of the ring homomorphism
    $$mathbbC[y_1,ldots,y_k]to mathbbC[x_1,ldots,x_n],quad y_imapsto p_i(x_1,ldots,x_n).$$
    (This enables us to compute $mathbbC[p_1,ldots,p_k]$).



    Example: Consider
    $$x_1x_2,x_3x_4,x_1x_3,x_2x_4inmathbbC[x_1,x_2,x_3,x_4].$$
    One obvious relation between them is
    $$(x_1x_2)(x_3x_4)=(x_1x_3)(x_2x_4).$$
    Thus, if
    $$varphi:mathbbC[y_1,y_2,y_3,y_4]tomathbbC[x_1,x_2,x_3,x_4]$$
    is the homomorphism defined by
    $$y_1mapsto x_1x_2,quad y_2mapsto x_3x_4,quad y_3mapsto x_1x_3,quad y_4mapsto x_2x_4,$$
    then $y_1y_2-y_3y_4inker varphi$. It is easy to see that $kervarphi$ is in fact generated by $y_1y_2-y_3y_4$, so
    $$mathbbC[x_1x_2,x_3x_4,x_1x_3,x_2x_4]congmathbbC[y_1,y_2,y_3,y_3]/(y_1y_2-y_3y_4).$$



    (I posted a similar question in mathematica.stackexchange.)










    share|cite|improve this question























      up vote
      3
      down vote

      favorite









      up vote
      3
      down vote

      favorite











      Let $p_1,ldots,p_kinmathbbC[x_1,ldots,x_n]$ be polynomials. Is there an algorithm to compute the ideal of relations between them?



      More precisely, to find a set of generators for the kernel of the ring homomorphism
      $$mathbbC[y_1,ldots,y_k]to mathbbC[x_1,ldots,x_n],quad y_imapsto p_i(x_1,ldots,x_n).$$
      (This enables us to compute $mathbbC[p_1,ldots,p_k]$).



      Example: Consider
      $$x_1x_2,x_3x_4,x_1x_3,x_2x_4inmathbbC[x_1,x_2,x_3,x_4].$$
      One obvious relation between them is
      $$(x_1x_2)(x_3x_4)=(x_1x_3)(x_2x_4).$$
      Thus, if
      $$varphi:mathbbC[y_1,y_2,y_3,y_4]tomathbbC[x_1,x_2,x_3,x_4]$$
      is the homomorphism defined by
      $$y_1mapsto x_1x_2,quad y_2mapsto x_3x_4,quad y_3mapsto x_1x_3,quad y_4mapsto x_2x_4,$$
      then $y_1y_2-y_3y_4inker varphi$. It is easy to see that $kervarphi$ is in fact generated by $y_1y_2-y_3y_4$, so
      $$mathbbC[x_1x_2,x_3x_4,x_1x_3,x_2x_4]congmathbbC[y_1,y_2,y_3,y_3]/(y_1y_2-y_3y_4).$$



      (I posted a similar question in mathematica.stackexchange.)










      share|cite|improve this question













      Let $p_1,ldots,p_kinmathbbC[x_1,ldots,x_n]$ be polynomials. Is there an algorithm to compute the ideal of relations between them?



      More precisely, to find a set of generators for the kernel of the ring homomorphism
      $$mathbbC[y_1,ldots,y_k]to mathbbC[x_1,ldots,x_n],quad y_imapsto p_i(x_1,ldots,x_n).$$
      (This enables us to compute $mathbbC[p_1,ldots,p_k]$).



      Example: Consider
      $$x_1x_2,x_3x_4,x_1x_3,x_2x_4inmathbbC[x_1,x_2,x_3,x_4].$$
      One obvious relation between them is
      $$(x_1x_2)(x_3x_4)=(x_1x_3)(x_2x_4).$$
      Thus, if
      $$varphi:mathbbC[y_1,y_2,y_3,y_4]tomathbbC[x_1,x_2,x_3,x_4]$$
      is the homomorphism defined by
      $$y_1mapsto x_1x_2,quad y_2mapsto x_3x_4,quad y_3mapsto x_1x_3,quad y_4mapsto x_2x_4,$$
      then $y_1y_2-y_3y_4inker varphi$. It is easy to see that $kervarphi$ is in fact generated by $y_1y_2-y_3y_4$, so
      $$mathbbC[x_1x_2,x_3x_4,x_1x_3,x_2x_4]congmathbbC[y_1,y_2,y_3,y_3]/(y_1y_2-y_3y_4).$$



      (I posted a similar question in mathematica.stackexchange.)







      abstract-algebra polynomials ring-theory algorithms polynomial-rings






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Sep 2 at 11:12









      Simon Parker

      1,4002917




      1,4002917

























          active

          oldest

          votes











          Your Answer




          StackExchange.ifUsing("editor", function ()
          return StackExchange.using("mathjaxEditing", function ()
          StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
          StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
          );
          );
          , "mathjax-editing");

          StackExchange.ready(function()
          var channelOptions =
          tags: "".split(" "),
          id: "69"
          ;
          initTagRenderer("".split(" "), "".split(" "), channelOptions);

          StackExchange.using("externalEditor", function()
          // Have to fire editor after snippets, if snippets enabled
          if (StackExchange.settings.snippets.snippetsEnabled)
          StackExchange.using("snippets", function()
          createEditor();
          );

          else
          createEditor();

          );

          function createEditor()
          StackExchange.prepareEditor(
          heartbeatType: 'answer',
          convertImagesToLinks: true,
          noModals: false,
          showLowRepImageUploadWarning: true,
          reputationToPostImages: 10,
          bindNavPrevention: true,
          postfix: "",
          noCode: true, onDemand: true,
          discardSelector: ".discard-answer"
          ,immediatelyShowMarkdownHelp:true
          );



          );













           

          draft saved


          draft discarded


















          StackExchange.ready(
          function ()
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2902610%2falgorithm-to-find-relations-between-polynomials%23new-answer', 'question_page');

          );

          Post as a guest



































          active

          oldest

          votes













          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes















           

          draft saved


          draft discarded















































           


          draft saved


          draft discarded














          StackExchange.ready(
          function ()
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2902610%2falgorithm-to-find-relations-between-polynomials%23new-answer', 'question_page');

          );

          Post as a guest













































































          這個網誌中的熱門文章

          How to combine Bézier curves to a surface?

          Carbon dioxide

          Why am i infinitely getting the same tweet with the Twitter Search API?