ד מצות עשה להחרים שבעה עממין, שנאמר "החרם תחרימם" (דברים כ,יז); וכל שבא לידו אחד מהן, ולא הרגו--הרי זה עובר בלא תעשה, שנאמר "לא תחייה, כל נשמה" (דברים כ,טז). וכבר אבדו, ואבד זכרם.
ה וכן מצות עשה לאבד זרע עמלק, שנאמר "תמחה את זכר עמלק" (דברים כה,יט); ומצות עשה לזכור תמיד מעשיו הרעים ואריבתו, כדי לעורר איבתו--שנאמר "זכור, את אשר עשה לך עמלק" (דברים כה,יז). מפי השמועה למדו, "זכור" בפה; "לא, תשכח" (דברים כה,יט) בלב, שאסור לשכוח איבתו ושנאתו.
The meforshim contrast the Rambam's formulation (Hil Melachim ch 5) of the mitzvah of fighting the seven nations of Canaan with the mitzvah of waging war against Amalek. In various Brisker-style seforim there are two diyukim that attract attention: 1) The Rambam writes that the seven nations of Canaan have vanished (based on Chazal's teaching that Sancherev mixed up the populations of all the nations so that their individual identity was lost); he omits this with respect to Amalek; 2) The Rambam writes that if you meet a member of the seven nations and fail to kill him you have violated a lav; he omits this detail in discussing Amalek.
The second point suggests that according to the Rambam, the mitzvah to wage war against Amalek is a chovas hatzibur, a communal mitzvah, while the mitzvah to kill the seven nations is a chovas hagavra on each individual (see this post at Havolim for more on this sugya). If you meet a member of Amalek there is nothing you as an individual need do, as the mitzvah to wage war can only be undertaken by the community as a whole.
The first point is addressed by the Rav in a footnote to his essay "Kol Dodi Dofek." He quotes his father as suggesting that Amalek is not a specific nation, but rather any nation which aims to destroy Klal Yisrael, any nation which adopts that ideology which we associate with Amalek, is to be treated as Amalek and destroyed. Amalek (unfortunately) continues to exist to the present day.
What confuses me is the rest of what the Rav says in that footnote (p 49-50 in Divrei Hagos v'Ha'aracha, if you want to look it up). He quotes his father as teaching that there are two dinim in the mitzvah of fighting Amalek: 1) A communal obligation to wage war, as discussed at the end of Parshas Beshalach; 2) An chiyuv on individuals to kill members of Amalek, as discussed at the end of Parshas Ki Teitzei. The chiddush that even a person who is an ideological-Amalekite even if not a genetic descendant of Amalek must be killed applies only to the first mitzvah, the communal chiyuv to wage war. However, the mitzvah of killing Amalek incumbent upon individuals applies only to true genetic Amalekites. "Milchama l'Hashem dor dor," the chiddush that war against Amalek is an eternal battle and hence applies to all who embrace the ideology of Amalek is a chiddush of Parshas Beshalach, the chovas hatzibur. It has no parallel in Parshas Ki Teitzei from where we learn the chiyuv incumbent upon individuals.
How does this approach square with the second diyuk we started with? Why does the Rambam not tell us (like he does with respect to the seven nations) that anyone who meets a member of Amalek and does not kill him has violated an issur, as there is a chovas hayachid to kill Amalek as well as a chovas hatzibur to wage war? Why would the Rambam in Sefer haMitzvos #248 categorize the destruction of Amalek as a chovas hatzibur when it also contains a chovas hayachid dimension as well? Why did Chazal choose the parsha that refers to the chiyuv of an individual to kill Amalek (Parshas Ki Teitzei) as the kri'ah for Parshas Zachor and not the parsha that refers to the communal chiyuv? (The Keren Orah at the beginning of Brachos writes that the reason why the d'oraysa chiyuv of reading parshas Amalek must be done b'tzibur is because the mitzvah of zecher is connected with the mitzvah of waging war which is a chovas hatzibur. Reading the parsha devoted to the chovas hayachid of killing Amalek undermines the point!) I'm sure I am missing something here...