# isTriangular -- Decides whether a polynomial system is triangular

## Synopsis

• Usage:
isTriangular F
isTriangular A
• Inputs:
• F, a list, of (Laurent) polynomial equations.
• A, a list, of matrices whose column vectors are the support of a system of (Laurent) polynomial equations
• Outputs:
• , a boolean asserting whether the polynomial system (or set of supports) is triangular

## Description

A polynomial system is triangular if, after a monomial change of coordinates, there is a proper subset of $k$ equations which involve only the first $k$ variables. This function checks whether a polynomial system (or set of supports) is triangular.

The function isTriangular accepts a list of polynomials forming a system.

 i1 : R=QQ[x,y]; i2 : F={3+x^2*y^2-(17/3)*x^4*y^4,2-x^2+5*y^2-13*x^2*y^2}; i3 : isTriangular F o3 = true

The function isTriangular also accepts a list of supports encoded as matrices.

 i4 : A = {matrix{{0,2,4},{0,2,4}},matrix{{0,0,2,2},{0,2,0,2}}}; i5 : isTriangular A o5 = true i6 : B = {matrix{{0,2,4},{0,2,3}},matrix{{0,1,0},{0,0,1}}}; i7 : isTriangular B o7 = false
{"ft_lang_label":"__label__en","ft_lang_prob":0.65345716,"math_prob":0.99777186,"size":750,"snap":"2022-40-2023-06","text_gpt3_token_len":259,"char_repetition_ratio":0.16756032,"word_repetition_ratio":0.0,"special_character_ratio":0.37733334,"punctuation_ratio":0.22916667,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99934006,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-02-01T16:32:57Z\",\"WARC-Record-ID\":\"<urn:uuid:1916aa72-4d55-4169-94f9-c7e96d190ad0>\",\"Content-Length\":\"6958\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7ef9275b-bc99-4b87-b59a-ec44c0c7f339>\",\"WARC-Concurrent-To\":\"<urn:uuid:b9ca109c-ff58-4591-9fdf-a9f86b41723a>\",\"WARC-IP-Address\":\"107.170.188.204\",\"WARC-Target-URI\":\"http://www2.macaulay2.com/Macaulay2/doc/Macaulay2-1.19/share/doc/Macaulay2/DecomposableSparseSystems/html/_is__Triangular.html\",\"WARC-Payload-Digest\":\"sha1:KQLWSWJACLWXBO3LUFIIQYIYUOF2UA4Q\",\"WARC-Block-Digest\":\"sha1:ALOUIHSR6FZPCYZBJ42R6L5NDLOVZKUV\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764499946.80_warc_CC-MAIN-20230201144459-20230201174459-00787.warc.gz\"}"}