# Big-oh for function of two variables

Is it true that $O(M^3 + NM^2) \, = \, O(M^3 + N)$, where $M$ and $N$ are variables of the function?

-
Set $M=0$; then $M^3 + NM^2$ vanishes, but $M^3 + N$ does not. –  user1119 Dec 16 '10 at 16:43
Technically, setting M=0 doesn't quite work, since the bound only has to hold for all M,N sufficiently large. –  Jeremy Hurwitz Dec 16 '10 at 17:50

It is true that $O(M^3+N)\subset O(M^3+NM^2)$, since for $M$ sufficiently large, $c(M^3+N)<c(M^3+NM^2)$. It's worth noting that for $M$ sufficiently close to $0$, $M^3+N>M^3+NM^2$.
For the reverse inclusion, consider $f(M,N)=M^3+NM^2$. Clearly $f\in O(M^3+NM^2)$. We now need to show that for all $c$ and arbitrarily large $M,N$ that $f(M,N)>c(M^3+N)$. We can do this by setting $M=\log N$. We then have $f(\log N,N)=\log^3 N + N\log^2 N$, which is clearly larger than $c(\log^3 N + N)$ for all sufficiently large $N$. So $f(M,N)\not\in O(M^3+N)$.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9841518998146057, "perplexity": 130.31201709731235}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440645378542.93/warc/CC-MAIN-20150827031618-00142-ip-10-171-96-226.ec2.internal.warc.gz"}