{"title":"多梯度麦考利对偶空间","authors":"Joseph Cummings, Jonathan D. Hauenstein","doi":"10.1142/s0219498825502469","DOIUrl":null,"url":null,"abstract":"<p>We describe an algorithm for computing Macaulay dual spaces for multi-graded ideals. For homogeneous ideals, the natural grading is inherited by the Macaulay dual space which has been leveraged to develop algorithms to compute the Macaulay dual space in each homogeneous degree. Our main theoretical result extends this idea to multi-graded Macaulay dual spaces inherited from multi-graded ideals. This natural duality allows ideal operations to be translated from homogeneous ideals to their corresponding operations on the multi-graded Macaulay dual spaces. In particular, we describe a linear operator with a right inverse for computing quotients by a multi-graded polynomial. By using a total ordering on the homogeneous components of the Macaulay dual space, we also describe how to recursively construct a basis for each component. Several examples are included to demonstrate this new approach.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Multi-graded Macaulay dual spaces\",\"authors\":\"Joseph Cummings, Jonathan D. Hauenstein\",\"doi\":\"10.1142/s0219498825502469\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We describe an algorithm for computing Macaulay dual spaces for multi-graded ideals. For homogeneous ideals, the natural grading is inherited by the Macaulay dual space which has been leveraged to develop algorithms to compute the Macaulay dual space in each homogeneous degree. Our main theoretical result extends this idea to multi-graded Macaulay dual spaces inherited from multi-graded ideals. This natural duality allows ideal operations to be translated from homogeneous ideals to their corresponding operations on the multi-graded Macaulay dual spaces. In particular, we describe a linear operator with a right inverse for computing quotients by a multi-graded polynomial. By using a total ordering on the homogeneous components of the Macaulay dual space, we also describe how to recursively construct a basis for each component. Several examples are included to demonstrate this new approach.</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-04-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1142/s0219498825502469\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1142/s0219498825502469","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
We describe an algorithm for computing Macaulay dual spaces for multi-graded ideals. For homogeneous ideals, the natural grading is inherited by the Macaulay dual space which has been leveraged to develop algorithms to compute the Macaulay dual space in each homogeneous degree. Our main theoretical result extends this idea to multi-graded Macaulay dual spaces inherited from multi-graded ideals. This natural duality allows ideal operations to be translated from homogeneous ideals to their corresponding operations on the multi-graded Macaulay dual spaces. In particular, we describe a linear operator with a right inverse for computing quotients by a multi-graded polynomial. By using a total ordering on the homogeneous components of the Macaulay dual space, we also describe how to recursively construct a basis for each component. Several examples are included to demonstrate this new approach.